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AEITHMETIC. 


ANALYTICAL   A^i  D   PRA 


BY 


CHARLES  DA  VIES,  LL.D. 

AUTHOU    OF    A     FULL     COURSE    OF     MATHEMATICS, 


REVISED     EDITION, 


PUBLISHED  BY  A.  S.  BARNES  &  CO., 

51    &   53   JOHN    STREET. 

SOLD    BY    BOOKSKLLERS,    GENERALLY,    THROUGHOUT    THE    UNITED    STATES. 
1858. 


r'lK 


ADVERTISEMENT 


Tub  uUeutiou  of  Teuchcrfc  is  respectiuUy  iuvited  to  the  Kkvibku  iLvi 

ItJNS  of 

§ai}iu'  ^rit|mttical  Beries 

FOR  SCHOOLS  AND  ACADEMIES. 

1.  DA  VIES'  PRIMARY  ARITHMETIC  AND  TABLE-BOOK. 

2.  DAVIES'  INTELLECTUAL  ARITHMETIC. 
8.  DAVIES'  SCHOOL  ARITHMETIC. 

4.  DAVIES'  UNIVERSITY  ARITHMETIC. 

5.  DAVIES'  PRACTICAL  MATHEMATICS. 


The  above  Works,  by  Charles  Davies,  LL.D.,  Author  ot  a  Complete 
Course  of  Mutliematic.>«,  are  designed  as  a  full  Course  of  Arithmetical  In- 
!*truction  necessary  for  the  practical  duties  of  business  life;  and  also  to 
prepare  the  Student  for  the  more  advanced  Series  of  Mathematics  by  the 
same  AutJior. 


EaUretf  Recording  to  Act  of  Congress,  In  the  year  one  tbousftiid  •Igbt  Luudred 

and  fifty-flve, 

By   CHARLES   DAVIES, 

la  Uie  (J'erk';!  OQloe  of  llio  District  Conrt  of  the  United  States  for  the  gouthent 

Dl&trict  of  New  York. 


PREFACE 


Arithmetic  embraces  the  science  of  numbers,  together  with  all  ttf 
lules  which  are  employed  in  applying  the  principles  of  this  science 
to  practical  purposes.  It  is  the  foundation  of  the  exact  and  mixed 
sciences,  and  the  first  subject,  in  a  well-arranged  course  of  instruc- 
tion, to  which  the  reasoning  powers  of  the  mind  are  directed.  Because 
of  its  great  practical  uses  and  applications,  it  has  become  the  guide 
and  daily  companion  of  the  mechanic  and  man  of  business.  Hence, 
a  full  and  accurate  knowledge  of  Arithmetic  is  one  of  the  most  im- 
portant elements  of  a  liberal  or  practical  education. 

Soon  after  the  publication,  in  1848,  of  the  last  edition  of  my  School 
Arithmetic,  it  occurred  to  ijie  that  the  interests  of  education  might  be 
promoted  by  preparing  a  full  analysis  of  the  science  of  mathematics, 
and  explaining  in  connection  the  most  improved  methods  of  teaching. 
The  results  of  that  undertaking  were  given  to  the  public  under  the 
title  of  "  Logic  and  Utility  of  Mathematics,  with  the  best  methods  of  in- 
struction explained  and  illustrated."  The  reception  of  that  work  by 
teachers,  and  by  the  public  generally,  is  a  strong  proof  of  the  deep  interest 
which  is  felt  in  any  effort,  however  humble,  which  may  be  made  to 
improve  our  systems  of  public  instruction. 

In  that  work  a  few  general  principles  are  laid  down  to  which  it  is 
ffupjjosed  all  the  operations  in  numbers  may  be  referred  : 

1st.  The  unit  1  is  regarded  as  the  base  of  every  number,  and  the 
consideration  of  it  as  the  first  step  in  the  aMalysis  of  every  question 
relating  to  numbers. 

"Zd.  Every  number  is  treated  as  a  collection  of  units,  or  as  made  up 
of  sets  of  such  collections,  each  collection  having  its  own  base,  which 
is  either  I,  or  some  number  derived  from  I. 

3^/  The  numbers  expressing  the  relation  between  the  different  units 
of  a  number  are  called  the  scale;  and  the  employment  of  this  term 
enables  us  to  generalize  the  laws  which  regulate  the  formation  of 
numbers. 

Ath.  By  employing  the  term  '* fractional  units  "  the  same  principles 
arc  made  applicable  to  fractional  numbers ,  for,  all  fractions  are  but 
Collections  of  fractional  units,  tlicse  units  having  a  known  relation  to  1 


rV  I'KKKAOE. 

lu  the  preparation  of  the  work,  two  objects  have  beeu  kept  con- 
Btuntly  in  view: 

\st.  To  make  it  educational ;  and, 

'2,(1.  To  make  it  practical. 

To  attain  these  ends,  the  following  plan  has  been  adopted : 

1.  To  introduce  every  new  idea  to  the  mind  of  the  pupil  by  a  sira- 
ole  question,  and  then  to  express  that  idea  in  general  terms  under  the 
form  of  a  definition. 

2.  When  a  sufficient  number  of  ideas  are  thus  fixed  in  the  mind, 
they  are  combined  to  form  the  basis  of  an  analysis ;  so  that  all  the 
principles  are  developed  by  analysis  in  their  proper  order. 

3.  An  entire  system  of  Mental  Arithmetic  has  been  carried  forward 
with  the  text,  by  means  of  a  series  of  connected  questions  placed  at 
the  bottom  of  each  page ;  and  if  these,  or  their  equivalents,  are  care- 
fully put  by  the  teacher,  the  pupil  will  understand  the  reasoning  in 
eveiy  process,  and  at  the  same  time  cultivate  the  powers  of  analysis' 
and  abstraction. 

4.  The  work  has  been  divided  into  sections,  each  containing  a  num- 
ber of  connected  principles ;  and  these  sections  constitute  a  series  of 
dependent  propositions  that  make  up  the  entire  system  of  principles 
and  rules  which  the  work  develops. 

Grettt  pains  have  beeu  taken  to  make  the  work  practioax  in  its 
general  character,  by  explaining  and  illustrating  the  various  applica- 
tions of  Arithmetic  in  the  transactions  of  business,  and  by  connecting 
as  closely  as  possible,  every  principle  or  rule,  with  all  the  applicationja 
which  belong  to  it. 

I  have  great  pleasure  in  acknowledging  my  obligations  to  many 
teachers  who  have  favored  me  with  valuable  suggestions  in  regard  to 
the  definitions,  rules,  and  meth«)d8  of  illustration,  in  the  previous  edi- 
tions. 1  hope  they  will  find  the  present  work  free  from  the  defects 
they  have  so  kindly  pointed  out. 

Much  more  than  a  general  acknowledgment  is  due  to  Mr.  D.  W. 
Fisu,  an  al>le  and  distinguished  teacher  of  Western  New  York,  who 
has  rendered  me  special  and  valuable  aid  in  the  preparation  of  this 
edition.  His  praciical  information  and  zealous  labors  have  given  ad- 
ditional value  to  many  parts  of  the  work. 

Flshkill  Lawoing,  February,  1856. 


CONTENTS 


?IR»T    riVK    RULES. 

Definition* 9— JO 

Notation  and  Numeration 10 — 22 

Addition  of  Simple  Numbers 22 — 30 

Applications  in  Addition 30 — 33 

Subtraction  of  Simple  Numbers 33 — 37 

Applications  in  Subtraction 37 — 42 

Multiplication  of  Simple  Numbers 42 — 50 

Factors 60—53 

Applications 63 — 56 

Division  of  Simple  Numbers , 56 — 6 1 

Fractions 61—64 

Long  Division 64 — 68 

Proof  of  Multiplication -. 68— «« 

Contractions  in  Multiplication 69 — 71 

Contractions  in  Division 71 — 74 

Applications  in  the  precodinir  Rules 74 — 79 

UNITED    STATES    MC^Ef. 

United  States  Money  defined 79 

Table  of  United  States  Money 79 

Numeration  of  United  States  Money 80 

Reduction  of  United  States  Money 81 — 83 

Addition  of  United  States  Money 83 — 85 

Subtraction  of  United  States  Money 85 — 87 

Multiplication  of  United  States  Money 87 — 91 

Division  of  United  States  Money 91 — 93 

Applications  in  the  Four  Rules. 93 — 96 

DENOMINATE    NUMBERS. 

English  Money 96—  97 

Reduction  of  Denominate  Numbers 97 —  99 

Linear  Measure 99 — 10  J 

Cloth  Measure 1 0 1— 102 

Lund  or  Squuie  Measure 102 — 1(V4 


fl  C0NTKNT6. 

Cubic  Measure  or  Measure  of  Volume 104 — 106 

Wine  or  Liquid  Measure. 106—108 

Ale  or  Beer  Measure. 108 — 101/ 

Dry  Measure 109—110 

Avoirdupois  Weight 1 10 — 1 1 1 

Troy  Weight 111—112 

Apothecaries'  Weight 112—1 14 

Measure  of  Time 114 — ^116 

Circular  Measure  or  Motion 116 

Miscellaneous  Table 117 

Miscellaneous  Examples 117 — 1\9 

Addition  of  iJenominate  Numbers H9 — 15^ 

Subtraction  of  Denominate  Numbers 124 — isfe 

Time  between  Dates 125 

Applications  in  Addition  and  Subtraction 126 — 12d, 

Multiplication  of  Denominate  Numbers 128 — 130 

Division  of  Denominate  Numbers 130 — 134 

Longitude  and  Time 134 

PKOPERTIE8    OF    NUMBKR8. 

Compt)8ite  and  Prime  Numbers 13.5 — 137 

Divisibility  of  Number^ 137 

Greatest  Common  Divisor 1 37 — 140 

Greatest  Common  Dividend 140 — 142 

Cancellation 142—145 

OF    COMMON    FRACTIONS. 

Definition  of,  and  First  Prinriples 14(5 — 149 

Of  the  different  kinds  of  Common  Fractions 149 — liiO 

Six  Fundamental  Propositions 150 — 154 

Reduction  of.  Common  Fractions 154 — 16 1 

Addition  of  Common  Fractious 161 — 162 

Subtraction  of  Common  Fractions 162 — 164 

Multiplication  of  Common  Fractions 164 — 168 

Division  of  Common  Fractions 168 — 172 

lieduction  of  Complex  Fractions 1 72 

Denominate  Fractions 1 73—  1 76 

Addition  and  Subtraction  of  Denominate  Fractions 176 — 178 

DUODEClMtLS. 

Defmitions  of,  &c .    178-  ISO 

Multiplication  uf  Duudcciiualsi IHO — l.s:.' 


CONTICNTS.  VII 

DECIMAL    rKACTlONS. 

Definition  of  Decimal  Fractions 1 82 — 183 

Decimal  Numeration — First  Principles 183 — 187 

Addition  of  Decimal  Fractions 187 — 19 1 

Subtraction  of  Decimal  Fractions , 191 — 193 

Multiplication  of  Decimal  Fractions 193 — 195 

Divis;on  of  Decimal  Fractions 195 — 197 

Applications  in  the  Four  Rules 197 — 198 

Denominate  Decimals 198 

Re<3uction  of  Denominate  Decimals 198 — 201 

ANALYSIS. 

Gmeral  Principles  and  Methods 201 — 213 

RATIO    AND    PROPORTION. 

jlatio  defined 213—214 

Proportion 214 — 210 

Simple  and  Compound  Ratio 216 — 218 

Single  Rule  of  Three 218—223 

Double  Rule  of  Three 223—228 

APPLICATIONS    TO    BUSINESS. 

Partnership 228 — 229 

Compound  Partnership 229 — 231 

Percentage 231—234 

Stock  Commission  and  Brokerage 234 — 237 

Profit  and  Loss 237—239 

Insurance 239-  -241 

Interest   241—247 

Partial  Payments 247 — 251 

Compound  Interest 251 — 253 

Discount 253 — 255 

Bank  Discount 255 — 257 

Equalion  of  Payments 257 — 260 

Assessing  Taxes 260 — 263 

Coins  and  Currency 263 — 264 

Reduction  of  Currencies 264 — 265 

Exchange 265 — 268 

Duties 268—271 

Aligation  Medial 27 1 — 272 

Alijiaiiua  Alternate     272 — ^276 


VIU  OO^'TICNTH. 

INVOLUTION. 

Definition  of,  dec 270 

EVOLUTION. 

Definition  of,  <fec 277 

Extraction  of  the  Square  Root «...   277—282 

Applications  in  Square  Root 282 — 285 

Extraction  of  the  Cube  Root 285—289 

Applications  in  Cube  Root 289-i290 

ARITHMETICAL    PROGRESSION.  j 

Definition  of,  (fee 290—201 

DiiFerent  Cases 291— 2k 

GEOMETRICAL    PROGRESSION.  \ 

Definition  of,  &c 294— 29^ 

Cases 295—291 

PROMISCUOUS    QUESTIONS. 

Questions  for  Practice 298  -  200 

MENSURATION- 

To  find  the  area  of  a  Triangle 303 

To  find  the  area  of  a  Square  Rectangle,  &c 302 

To  find  the  area  of  a  Trapezoid 304 

To  find  the  circumference  and  diameter  of  a  Circle 304 

To  find  the  area  of  a  Circle 305 

To  find  the  surface  of  a  Sphere 305 

To  find  the  contents  of  a  Sphere 305 

To  find  the  convex  surface  of  a  Prism 306 

To  find  the  contents  of  a  Prism. 306 

To  find  the  convex  surface  of  a  Cylinder 307 

To  find  the  contents  of  a  Cylinder 307 

To  find  the  contents  of  a  Pyramid 308 

To  find  the  contents  of  a  Cone 308 

GAUGIWrO. 

Rules  for  Gauging 309 

APPENDIX. 

Forms   relatmg  to  Business  in  General 310 — 313 


ARITHMETIC. 


DEFINITIONS. 

1.  A  SINGLE  THING  is  Called  one  or  a  unit, 

2.  A  NUMBER  is  a  unit,  or  a  collection  of  units.  The  unit 
is  called  the  base  of  the  collection.  The  primary  base  of 
every  number  is  the  unit  one. 

3.  Each  of  the  words,  or  terms,  one,  two,  three,  four,  &c., 
denotes  how  many  things  are  taken.  These  terms  are  gene- 
rally called  numbers ;  though,  in  fact,  they  are  but  the 
names  of  numbers. 

4.  The  term,  one,  has  no  reference  to  the  kind  of  thing  to 
which  it  is  applied  :  and  is  called  an  abstract  unit. 

5.  The  term,  one  foot^  refers  to  a  single  foot,  and  is  callcil 
a  concrete  or  de?tominate  unit. 

6.  An  abstract  number  is  one  whose  unit  is  abstract  :  thus, 
three,  four,  six,  &zc.,  are  abstract  numbers. 

7.  A  concrete  or  denominate  number,  is  one  whose  unit  is 
concrete  or  denominate :  thus,  three  feet,  four  dollars,  five 
pounds,  are  denominate  numbers. 

1.  What  is  a  single  thing  called  ! 

2.  What  is  a  number!  What  is  the  unit  called"?  W^hat  is  the 
primary  base  of  every  number  1 

3.  What  does  each  of  the  words,  one,  two,  three,  denote  1  Wftat  are 
these  words  generally  called  I     What  are  they,  in  facti 

4.  Has  the  term  one  any  reference  to  the  thing  to  which  it  may  be 
applied  ^     What  is  it  called  1 

5.  What  does  the  term,  one  foot,  refer  to !     What  is  it  called  ? 

G.  What  is  an  abstract  number  1  Give  examples  of  abstract  nuiUf 
bers. 

7.  What  is  a  concrete  or  denominate  number  1  aive  examples  of 
deiiomiuate  numbers. 


10  DICFITs'ITlONS. 

8.  A  Simple  number  is  a  single  collection  of  units,  whether 
abstract  or  denominate. 

9.  Q,UANTiTY  is  anything  which  can  be  measured  by  a  unit. 

'     10,   Science  treats  of  the  properties  and  relations  of  things  : 
Art  is  the  practical  application  of  the  principles  of  Science. 

11.  Arithmetic  treats  of  numbers.  It  is  a  sa> wee  when 
it  makes  known  the  properties  and  relations  of  numbers  ;  and 
an  art^  when  it  applies  principles  of  science  to  practical  pur- 
poses. 

12.  A  Proposition  is  something  to  be  done,  or  demonstrated. 

13.  An  Analysis  is  an  examination  of  the  separate  parts 
of  a  proposition. 

14.  An  Operation  is  the  act  of  doing  something  with 
numbers.  The  number  obtained  by  an  operation  is  called  a 
re  .suit,  or  answer. 

16.  A  Rule  is  a  direction  for  performing  an  operation,  and 
may  be  deduced  either  from  an  analysis  or  a  demonstration. 

16.  There  are  five  fundamental  processes  of  Arithmetic : 
Notation  and  Numeration,  Addition,  Subtraction,  Multiplica- 
tion and  Division. 

EXPRESSING  NUMBERS. 

17.  There  are  three  methods  of  expressing  numbers  : 

1st.  By  words,  or  common  language  ; 

2d.    By  letters,  called  the  Roman  method ; 

3d.    By  figures,  called  the  Arabic  method. 


8.  What  is  a  simple  number  1 

9.  What  is  quantity  1 

10.  Of  what  does  Science  treat  ?     What  is  Art  1 

11.  Of  what  does  Arithmetic  treat  !     When  is  it  a  science  ?     When 
ail  art  1 

12.  What  is  a  Proposition  T 

13.  What  is  an  Analysis  "? 

14.  What  is  an  Operation  \     What  is  the  number  obtained  called  1 

15.  What  is  a  Rule  1      How  may  it  be  deduced  ? 

16.  How  many  fundamental  rules  are  there  1     What  are  they  1 

17.  How  many  methods  are  there  of  expressing  numbers?     Wliat 
aie  they  * 


11 


18.  A  single  thing  is  called    -         -         -         .  One. 

Two. 
Three 
Pour. 
Five. 
Six. 
Seven. 
Eight. 
Nine. 
Ten. 

&iC. 

Each  of  the  words,  one,  two,  three,  four,  jive,  six,  &c., 
denotes  how  many  things  are  taken  in  the  collection. 

NOTATION. 

19.  Notation  is  the  method  of  expressing  nnmbers  either 
by  letters  or  figures.  The  method  by  letters,  is  called  Rar/tan 
Notation  ;  the  method  by  figures  is  called  Arabic  Notation. 

ROMAN  NOTATION. 

20.  In  the  Roman  Notation,  seven  capital  letters  are  used, 
viz  :  I,  stands  ior  one  ;  V,  for  Jive  ;  X,  ibr  ten  ;  L,  tor  fifty  ; 
C,  hv  one  hundred;  D,  ibr  five  hundred;  and  M,  for  07ie 
thousand.  All  other  numbers  are  expressed  by  combining 
the  letters  according  to  the  following 


^  NOTATION. 

BY  WORDS. 

A  sing 

le  thing  is  called    -         -         - 

One 

and  one  more 

Two 

and  one  more 

Three 

and  one  more 

Four 

and  one  ihore 

Five 

and  one  more 

Six 

and  one  more 

Seven 

and  one  more 

Eight 

and  one  more 

Nine 

and  one  more 

&.C. 

Ic. 

ROMAN  TABLE. 

i.     .     . 

-     One. 

LXX.    - 

-     Seventy. 

II.  -     - 

-     Two. 

LXXX. 

-     Eighty. 

III.-     - 

-     Three. 

XC.      - 

-     Ninety. 

IV.  -     - 

-     Four. 

C.    .     - 

-     One  hundred. 

V.   -     - 

-     Five. 

CC.      - 

-     Two  hundred. 

VI  -     - 

-     Six. 

CCC.  - 

-     Three  hundred. 

VII.      - 

-     Seven. 

cccc 

-     Four  hundred. 

VIII.    - 

-     Eight. 

D.    -     - 

-     Five  hundred. 

IX.-     - 

-     Nine. 

DC.      - 

-     Six  hundred. 

X.    -     - 

-     Ten. 

DCC.   . 

-     Seven  hundred. 

XX.      . 

-     Twenty. 

DCCC. 

-     Eight  hundred. 

XXX.    - 

-     Thirty. 

DCCCC. 

-     Nine  hundred. 

XL.      . 

-     Forty. 

M.  -     - 

■     One  thousand. 

L.    -     - 

-     Fifty. 

MD.     - 

Fifteen  hundred 

LX       - 

Sixty. 

MM.    . 

.     Two  thousand. 

12  NOTATiUM. 

Note. — The  principles  of  this  Notation  arc  these 

1.  Every  time  a  letter  is  repeated,  the  number  which  it  denotefc 
is  also  repeated. 

2.  If  a  letter  denoting  a  less  number  is  written  on  the  right  of 
one  denoting  a  greater ^  their  sum  will  be  the  number  expressed, 

3.  If  a  letter  denoting  a  less  number  is  written  on  the  left  of 
one  denoting  a  greater,  their  difference  will  be  the  number  ex. 
pressed. 

EXAMPLES  IN  ROMAN  NOTATION. 

Express  the  following  numbers  by  letters  : 

1.  Eleven. 

2.  Fifteen. 

3.  Nineteen. 

4.  Twenty-nine. 
6.   Thirty-five. 

6.  Forty-seven. 

7.  Ninety-nine. 

8.  One  hundred  and  sixty. 

9  Four  hundred  and  tbrty-one. 

10.  Five  hundred  and  sixty-nine. 

11.  One  thousand  one  hundred  and  six. 
\  2.  Two  thousand  and  twenty-five. 

13.  Six  hundred  and  ninety-nine. 

14.  One  thousand  nine  hundred  and  twenty-five. 

15.  Two  thousand  fiix  hundred  and  eighty. 

16.  Four  thousand  nine  hundred  and  sixty-five. 

17.  Two  thousand  seven  hundred  and  ninety -one. 

18.  One  thousand  nine  hundred  and  sixteen. 

19.  Two  thousand  six  hundred  and  forty-one. 

20.  One  thousand  three  hundred  and  forty-two. 

19.  What  is  Notation  {  What  is  the  method  by  letters  called  1  What 
is  the  method  by  figures  called  1 

20.  How  many  letters  are  used  in  the  Roman  notation  1  Which  are 
they  1     What  does  each  stand  for  1 

Note. — What  takes  place  when  a  letter  is  repeated  1  If  a  letter  de- 
noting a  less  number  be  placed  on  the  right  of  one  denoting  a  greater, 
now  are  they  read  1  If  the  letter  denoting  the  less  number  be  written 
on  the  left,  how  are  they  read  ^ 

21.  What  is  Arabic  Notation  1  How  many  figures  are  used  I  What 
do  they  form  1  Name  the  figures.  How  many  things  does  1  express  ? 
How  many  things  does  2  express  ]  How  many  units  in  3  ]  In  4  !  In 
6  ]  In  9  1  In  8  ?  What  does  0  express  ^  What  are  the  other  figures 
calkd  I 


^OTAT'UN.  It) 

ARABIC  NOTATION. 

21.  Arabic  Notation  is  the  method  of  expressing  uumberg 
by  figures.  Ten  figures  are  used,  and  they  Ibrm  the  alpiiaJbet 
of  Ute  Arabic  Notation. 

Thev  are,       0  called  zero,  cipher,  or  Naught, 

1  -  .  One. 

2  ■  -  -  Two. 

3  -  -  Three. 
■    4             -             -             Four. 

6  -  -  Five. 

6  -  -  Six. 

7  -  -  Seven. 

8  -  -  Eight. 

9  -  -  Nine. 

1  expresses  a  single  thing,  or  the  unit  oi'  a  number. 


two  things  -  -  or  two  units, 

three  things  -  -  or  three  units, 

four  things  -  -  or  four  units, 

five  things  -  •  or  five  units, 

six  things  -  -  or  six  uiiits. 

seven  things  -  -  or  seven  units, 

eight  things  -  -  or  eight  units 

nine  thinjjs  -  -  or  nine  units. 


The  cipher,  0,  is  used  to  denote  the  absence  of  a  thing 
Thus,  to  express  that  there  are  no  apples  in  a  basket,  we 
write,  the  number  of  apples  is  0.  The  nine  other  figures  are 
called  significant  figures,  or  Digits. 

22.  We  have  no  single  figure  for   the  number  ten.     We 
therefore  combine  the  figures  already  known.     This  we  do  by 
writing  0  on  the  right  hand  of  1,  thus  : 
10,  which  is  read  ten. 

This  10  is  equal  to  ten  of  the  units  expressed  by  1.  It  is, 
however,  but  a  si?igle  ten,  and  may  be  regarded  as  a  unit 
the  value  of  which  is  ten  times  as  great  as  the  unit  I.  It  is 
called  a  unit  of  the  second  order. 


22.  Have  we  a  separate  character  for  ten  I  How  do  we  express  ten  1 
To  how  many  units  I  is  ten  equal  1  May  we  consider  it  a  single  unit' 
Of  wiiai  order  :' 


14  NOTATION. 

23.  When  two  figures  are  written  by  th  3  side  of  each  other, 
the  one  on  the  rigiit  is  in  the  'place  of  units,  and  the  other  in 
the  flace  of  tens,  or  of  units  oj  the  second  Qrder.  Each  unit 
of  the  second  order  is  equal  to  ten  units  of  the  first  order. 

When  units  simply  are  named,  units  of  the  first  order  a^t 
always  meant. 

Two  tens,  or  two  units  of  the  second  order,  are  written  26 

Three  tens,  or  three  units  of  the  second  order,  are  written  30 

Four  tens,  or  four  units  of  the  second  order,  are  written  40 

Five  tens,  or  five  units  of  the  second  order,  are  written  50 

Six  tens,  or  six  units  of  tlie  second  order,  are  written  60 

Seven  tens,  or  seven  units  of  the  second  order,  are  written  70 

Eight  tens   or  eight  units  of  the  second  order,  are  written  80 

Nine  tens,  or  nine  units  of  the  second  order,  are  written  90 

These   figures   are  read,  twenty,  thirty,  forty,  fifty,  sixty, 
seventy,  eighty,  ninety. 

The  intermediate  numbers  between  10  and  20,  between  20 
and  30,  &c.,  may  be  readily  expressed  by  considering   their 
tens  and  units.      For  example,  the  number  twelve  is  made 
up  of  one  ten  and  two  units.     It  must  therefore  be  written 
by  setting  1  in  the  place  of  tens,  and  2  in  the  place  of  units  . 
thus,      ......  12 

Eighteen  has  1  ten  and  8  units,  and  is  written       -  -  18 

Twenty-five  has  2  tens  and  5  units,  and  is  written  -         25 

Thirty-seven  has  3  tens  and  7  units,  and  is  written  -         37 

Fifly-four  has  5  tens  and  4  units,  and  is  written    -  -         54 

Hence,  any  number  greater  than  nine,  and  less  than  one 
hundred,  may  be  expressed  by  two  figures. 

24.  In  order  to  express  ten  units  of  the  second  order,  or 
one  hundred,  we  form  a  new  combination. 

It  is  done  thus,  -  -  -  100 

by  writing  two  ciphers  on  the  right  of  1.     This  number  is 
read,  one  hundred. 


23.  When  two  figures  are  written  by  the  side  of  each  other,  what  \» 
the  place  on  the  right  called  !  The  place  on  the  left  !  When  units 
simply  are  named,  what  units  are  meant  1  How  many  units  of  the 
second  order  in  20  !  In  30  \  In  40 1  In  50  \  In  60  !  In  70  1  lu 
80  ?  In  90  I  Of  what  is  the  number  12  made  up  I  Also  18,  2o,  37 
54  1     What  numbers  may  be  expressed  by  two  figures  1 


NOTATION.  15 

Now  this  one  hundred  expresses  10  units  of  the  second 
m-der,  or  100  units  of  the  first  order.  The  one  hundred  is  but 
an  individual  hundred,  and,  in  this  light,  may  be  regarded 
as  a  unit  of  the  tJdrd  order. 

We  can  now  express  any  number  less  than  one  thousand. 

For    example,    in    the  number   three    hundred    and 
Beventy-five,  there  are  5  units,  7  tens,  and  3   hundreds,     g    *  -^ 
Write,  therefore,  5  units  of  the  tirst  order,  7  units  of  the    J   |    § 
second  order,    and   3  of  the   third  ]  and  read  from  the     3    7    5 
right,  units^  tensj  hundreds. 

In  the  number  eight  hundred  and  ninety-nine,  there  ^  ^  ^ 
are  9  units  of  the  first  order,  9  of  the  second,  and  8  of  a  g  "S 
the  third  ;  and  is  read,  units,  tens,  hundreds.  8   9    9 

In  the  number  four  hundred  and  six,  there  are  6  units  «  „;  2 
of  the  first  order,  0  of  the  second,  and  4  of  the  third.  5    §    S 

The  right  liand  figure  always  expresses  units 
of  the  first  order;  the  second,  units  of  the  second  order;  and 
the  third,  units  of  the  third  order. 

25.  To  express  ten  units  of  the  third  order,  or  one  thous 
and,  we  form  a  new  combination  by  writing  three  ciphers  on 
the  right  of  1  ;  thus,  -  -  -  1000 

Now,  this  is  but  one  single  thousand,  and  may  be  regarded 
as  a  miit  of  the  fourth  order. 

Thus,  we  may  form  as  many  orders  of  units  as  we  please  : 
a  single  unit  of  the  first  order  is  expressed  by  I, 

a  unit  of  the  second  order  by  1  and  0  ;  thus,  10, 

a  unit  of  the  third  order  by  1  and  two  O's  ;  100, 

a  unit  of  the  fourth  order  by  1  and  three  O's ;  1000, 

a  unit  of  the  fifth  order  by  1  and  four  O's ;  10000  ; 

and  so  on,  ibr  units  of  higher  orders  : 


24.  How  do  you  write  one  hundred  1  To  how  many  units  of  the 
second  order  is  it  equal  1  To  how  many  of  the  first  order  1  May  it  be 
considered  a  single  unit  !  Of  what  order  is  it  !  How  many  units  of 
the  third  order  in  200  !  In  300  !  In  400?  In  5001  In  6001  Of 
what  is  the  number  375  composed  1  The  number  899  \  The  number 
406  !  What  numbers  may  be  expressed  by  three  figures  ■  Wha» 
urdcl  uf  units  will'each  figure  express  • 


1 6  NOTATION. 

26,  Therefore, 

1st.  The  same  jig \ ire  expresses  different  units  according  to 
the  place  which  it  occupies  : 

2d.  Units  of  the  first  order  occupy  the  place  on  the  right ; 
units  of  the  second  order,  the  second  place  ;  units  of  the  third 
order,  the  third  place  ;  and  so  on  for  places  still  to  the  left : 

3d.  Ten  units  of  the  first  order  make  one  of  the  second ; 
ten  of  the  second,  one  of  the  third  ;  ten  of  the  third,  one  of  the 
fourth  ;  and  so  on  for  the  higher  orders  : 

4th.  When  figures  are  written  hy  the  side  of  each  other, 
ten  units  in  any  one  place  make  one  unit  of  the  pdace  next 
to  the  left. 

EXAMPLES    IN    WRITING    THE    ORDERS    OF    UNITS. 

1.  Write  3  tens. 

2.  Write  8  units  of  the  second  order. 

3.  Write  9  units  of  the  first  order. 

4.  Write  4  units  of  the  first  order,  5  of  the  second,  6  of  the 
third,  and  8  of  the  fourth. 

5.  Write  9  units  of  the  fifth  order,  none  of  the  fourth,  8  of 
the  third,  7  of  the  second,  and  6  of  the  first.       Ans.  90876. 

6.  Write  one  unit  of  the  sixth  order,  5  of  the  fiftli,  4  of  the 
fourth,  9  of  the  third,  7  of  the  second,  and  0  of  the  first 

Ans. 

7.  Write  4  units  of  the  eleventh  order. 

8.  Write  forty  units  of  the  second  order. 

9.  Write  60  units  of  the  third  order,  with  four  of  the  2d, 
and  5  of  the  first. 

10.  Write  6  units  of  tl>e  4th  order,  with  8  of  the  3d, 
4  of  the  1st. 

25.  To  what  are  ten  snits  of  the  third  order  equal  1  How  do  you 
write  it  1  How  is  a  single  unit  of  the  fir,si  order  written  1  How  do 
you  write  a  unit  of  the  second  order  1  One  of  the  third  1  One  of  the 
fourth!     One  of  the  fifth ! 

26.  On  what  does  the  unit  of  a  figure  depend  1  What  is  the  unit  Oi 
the  first  place  on  the  right  1  What  is  the  unit  of  the  second  place  1 
What  is  the  unit  of  the  third  place  1  Of  the  fourth  ?  Of  the  fifth  1 
Sixth  1  How  many  units  of  the  first  order  make  one  of  the  second  ] 
How  many  of  the  second  one  of  the  third  1  How  many  of  the  third  one 
of  the  fourtii,  Ac.  When  figures  are  written  by  the  side  of  each  other, 
how  many   units   of  any  place  make  one  utjit   of  the'  place  next  to  the 


NUMERATION.  17 

11.  Write  9  units  of  tiie  5th  order,  0  of  the  4th,  8  of  the 
Sd,  1  of  the  2d,  and  3  of  the  1st. 

12.  Write  7  units  of  the  6th  order,  8  of  the  5th,  0  of  the 
4tli,  5  of  the  3d,  7  of  the  2d,  and  1  of  the  11th. 

13.  Write  9  units  of  tlie  7th  order,  0  of  the  6th,  2  of  the 
5th,  3  of  the  4th,  9  of  the  3d.  2  of  the  2d,  and  9  of  the  1st. 

14.  Write  8  units  of  the  8th  order,  6  of  the  7th,  9  of  tho 
0th,  8  of  the  5th,  1  of  the  4th,  0  of  the  3d,  2  of  the  2d,  and 
8  of  the  1st. 

15.  Write  1  unit  of  the  9th  order,  6  of  the  8th,  9  of  the 
7th,  7  of  the  6th,  6  of  the  5th,  5  of  the  4th,  4  of  the  3d,  3  of 
the  2d,  and  2  of  the  1st. 

16.  Write  8  units  of  the  10th  order,  0  of  the  9th,  0  of  the 
8th,  0  of  the  7th,  9  of  the  6th,  8  of  the  5th,  0  of  the  4th,  3 
of  the  3d,  2  of  the  2d,  and  0  of  the  1st. 

17.  Write  7  units  of  the  ninth  order,  with  6  of  the  7th,  9 
of  the  third,  8  of  the  2d,  and  9  of  the  1st. 

18.  Write  6  units  of  8th  order,  with  9  of  the  6th,  4  of  the 
6th,  2  of  the  3d,  and  1  of  the  1st. 

19.  Write  14  units  of  the  12th  order,  with  9  of  the  10th, 
6  of  the  8th,  7  of  the  6th,  6  of  the  5th,  6  of  the  3d,  and  3 
of  the  first. 

20.  Write  13  units  of  the  13th  order,  8  of  the  12th,  7  of 
the  9th,  6  of  the  8th,  9  of  the  7  th,  7  of  the  6th,  3  of  the  4th, 
and  9  of  the  first. 

21.  Write  9  units  of  the  18th  order,  7  of  the  16th,  4  of  the 
I5th,  8  of  the  12th,  3  of  the  11th,  2  of  the  10th,  1  of  the 
9th,  0  of  the  e^h,  6  of  the  7th,  2  of  the  third,  and  1  of  the  1st. 

NUMERATION. 

27.  Numeration  is  the  art  of  reading  correctly  any  num- 
Der  expressed  by  figures  or  letters. 

The  pupil  has  already  been  taught  to  read  all  numbers  from 
jne  to  one  thousand.  The  Numeration  Table  will  teach  him 
to  read  any  number  whatever  ;  or,  to  express  numbers  in  words. 


27.  What  is  Numeration  1  What  is  the  unit  of  the  first  period ' 
What  is  the  unit  of  the  second  1  Of  the  third  1  Of  the  fourth  I  Of 
the  fifth  1  Sixth"?  Seventh  I  Eiglith  1  Give  the  rule  fur  reading 
iiuuiheii. 


18 


NUMiaiATION. 


NUMERATION  TABLE. 


Glh  Period.    5th  I'eriod.     4th  Period. 
Quadrillions.  Trillions.       Billions. 


3d  Period.     2d  Peri<  d.     1st  PenotL 

Millions.     Thousands.         Units. 


^    O    3 


5  7 
2  0, 


S3      •      • 

S     CO 

gl  • 

^      .2 


3  S-c 


8 
9  1 

4  0 

2  8 

3  2 


o  ^ 


•-=!        '5   «  .2 


ffiHPQ 


3,       8  4  2, 


^.2 


o 

3    CD  •- 


•73 


4  3, 

4  8, 

4  5, 

4  9, 

0  2, 

0  0, 


7  6, 

3  6, 

4  1, 
6  8, 


5    w 

2-^    • 

EI!  <^ 

^     02     3 

5  §^ 


8  2, 

1  2  3, 
0  0  0, 

2  1  0, 
0  0  0, 
2  8  9, 

3 
4 
0 
1 
2 
7 


2  9  7, 

3  1   9i 


5 


c 

6 

7  5 
7  9 
2  3 
0  1 


2   1 

2  2 


Notes. — 1.  Numbers  expressed  by  more  than  three  figures  are 
written  and  read  by  periods,  as  shown  in  the  above  tabje. 

2.  Each  period  always  contains  three  figures,  except  the  last 
which  may  contain  either  one,  two,  or  three  figures. 

3.  The  unit  of  the  first,  or  right-hand  period,  is  1 .;  of  the  second 
period,  1  thousand  ;  of  the  3d,  1  million ;  of  the  fourth,  1  bil- 
lion ]  and  so,  for  periods,  still  to  the  left. 

4.  To  quadrillions   succeed   quintillions,   sextillions,  seplilhouE, 


5.  The  pupil  should  be  required  to  commit,  thfj'oughly,  the 
names  of  the  periods,  so  as  to  repeat  them  in  their  regular  (irder 
fiom  left  to  riglit,  as  "well  as  from  right  to  left. 


NL'MEItA'l'tOM. 


li» 


RULE    FOR    READING    NUMBERS. 

I.  Divide   the  number   into  periods  of  three  figures  each^ 
bee/ in /I  in;/  at  the  riyltt  hand. 

II.  Name  the  order  of  each  figure^  beginning  at    the  right 
hand 

III.  Then,  beginning  at  the   left  hand,  read  each  period  iw 
if  it  stood  alone,  naming  its  unit. 

EXAMPLES    IN    READING    NUMBER.S. 

28.   Lei  the  pupil  point  ofi'aiid  read  the  tbllowiiig  numbei-s 
— then  write  them  in  words. 


1. 

67 

7. 

6124076 

13.      804321049 

2. 

125 

8. 

8073405 

14.    90067236708 

3. 

6256 

9. 

261^40123 

15.   870432697082 

4. 

4697 

10. 

9602316 

16.  1704291672301 

5. 

23697 

11. 

87000032 

17.  3409672103604 

6. 

412304 

12.     1987004086 

18.  49701342641714 

19. 

8760218760541 

23.     9080620359704567 

20. 

904326170365 

24.     9806071234560078 

21. 

30267821040291 

25.    30621890367081263 

22. 

90762038 

0467026 

26.   350673123051672607 

Note. — Let  each  of  the  above  examples,  after  being  written  on 
tiie  black  board,  be  analyzed  as  a  clays  exercise ;  thus  : 

Ex.  1.  How  many  tens  in  67  ?     How  many  units  over  ? 

2.  In  125,  how  many  hundreds  in  the  hundreds  place?  How 
many  tens  in  the  teiis  place  ?  How  many  units  in  the  units 
place  ?     How  many  tens  in  the  number  ? 

3.  In  6256,  how  many  thousands  in  the  thousands  place  ?  How 
many  hundreds  in  the  hundreds  place  ?  How  many  tens  in  the 
tens  place  ?     How  many  units  in  the  units  place? 

4.  How  many  thousands  in  the  number  4697  ?  How  manj 
hundreds?     How  many  tens  ?     How  many  units? 

5.  How  many  thousands  in  the  number  23697  ?  How  many 
hundreds  ?     How  many  tens  ?     How  many  uiuts  ? 

6.  How  many  hundreds  of  thousands  in  412304?  How  many 
ten  thousands  ?  How  many  thousands  ?  How  many  hundreds  ? 
How  many  tens  ?     How  many  units  ? 


US.  Name  the  unit!^  uf  each  order  in  ewduyh  Ml     In    10^     In   IC 
In  W^     Jiivo  tht  rule  for  wfitinii  niuiihcrsi 


20  NUMKIiATlON. 


KULE    Foil    WRlTliNG    NUMBERS     OR    NOTAllON. 

I.  Begin  at  the  lejt  hand  and  write  each  period  in  order ^  as 
if  it  were  a  period  of  units. 

II.  When  the  number  in  any  period,  except  the  left  kana 
period,  is  expressed  by  less  than  three  figures,  prefix  07ie  or  two 
ciphers  ;  and  when  a  vacant  j^criod  occurs,  Jill  it  with  ciphers, 

EXAMPLES    IN    NOTATION. 

29.  Express  the  following  numbers  in  figures  : 

1.  One  hundred  and  five. 

2.  Three  hundred  and  two. 

3.  Five  hundred  and  nineteen. 

4.  One  thousand  and  four. 

5.  Eight  thousand  seven  hundred  and  one. 

6.  Eorty  thousand  four  hundred  and  six. 

7.  Fifty-eight  thousand  and  sixty-one. 

8.  Ninety-nine  thousand  nine  hundred  and  ninety-nine. 

9.  Four  hundred  and  six  thousands  and  forty-nine. 

10.  Six  hundred  and  forty-one  thousand,  seven  hundred  and 
twenty-one. 

11.  One  rnilhon,  four  hundred  and  twenty-one  thousands, 
six  hundred  and  two. 

12.  Nine  millions,  six  hundred  and  twenty-one  thousands, 
and  sixteen. 

13.  Ninety -four  milhons,  eight  hundred  and  seven  thous- 
ands, four  hundred  and  nine. 

14.  Four  billions,  three  hundred  and  six  thousands,  nine 
hundred  and  nine. 

15.  Forty-nine  billions,  nine  hundred  and  forty-nine  thous- 
ands, and  sixty-five. 

16.  Nine  hundred  and  ninety  billions,  nine  hundred  and 
ninety-nine  millions,  nine  hundred  and  ninety  thousands,  nine 
hundred  and  ninety-nine. 

17.  Four  hundred  and  nine  billions,  two  hundred  and  nine 
lliousands,  one  hundred  and  six. 

18.  Six  hundred  and  forty-five  billio.is,  two  hundred  and 
Bixtytnine  miUions,  eight  luindicd  and  lilty-nine  thousaiid>^ 
nine  huutlre<l  ami  t^ix 


M'MK'JATrOxV.  2! 

19.  Forty-seven  millions,  two  hundred  and  four  thousands., 
eight  hundred  and  fifty-one. 

20.  Six  quadrillions,  forty-nine  trillions,  seventy-two  bil- 
lions, four  hundred  and  seven  thousands,  eight  hundred  and 
bixty-one. 

21.  Eight  hundred  and  ninety-nine  quadrillions,  four  hun- 
dred and  sixty  trillions,  eight  hundred  and  fifty  billions,  two 
Aundred  millions,  five  hundred  and  six  thousands,  four  hun- 
dred and  ninety-nine. 

22.  Fifty-nine  trillions,  fifty-nine  billions,  fifty-nine  millions, 
fifty-nine  thousands,  nine  hundred   and  fifty-mne. 

23.  Eleven  thousands,  eleven  hundred  and  eleven. 

24.  Nine  billions  and  sixty-five. 

25.  Write  three  hundred  and  four  trillions,  one  million, 
three  hundred  and  twenty-one  thousands,  nine  hundred  and 
forty-one. 

26.  Write  nine  trillions,  six  hundred  and  forty  billions, 
with  7  units  of  the  ninth  order,  6  of  the  seventh  order,  8  of 
the  fifth,  2  of  the  third,  1  of  the  second,  and  3  of  the  first. 

27.  Write  three  hundred  and  five  trillions,  one  hundred  and 
four  billions,  one  million,  with  4  units  of  the  fifth  order,  5  of 
the  fourth,  7  of  the  second,  and  4  of  the  first. 

28.  Write  three  hundred  and  one  billions,  six  millions,  four 
thousands,  with  8  units  of  the  fourteenth  order,  6  of  the 
third,  and  two  of  the  second. 

29.  Write  nine  hundred  and  four  trillions  six  hundred  and 
six,  with  4  units  of  the  eighteenth  order,  five  of  the  sixteenth, 
four  of  the  twelfth,  seven  of  the  ninth,  and  6  of  the  fifth. 

30.  Write  sixty-seven  quadrillions,  six  hundred  and  forty- 
one  billions,  eight  hundred  and  four  millions,  six  hundred  and 
forty-four. 

31.  Write  eight  hundred  and  three  quintillions,  sixty-nine 
bilhons,  four  hundred  and  forty  millions,  nine  hundred  thous- 
and and  three. 

32.  Write  one  hundred  and  fifty-nine  sextillions,  four  hun- 
dred and  five  billions,  two  hundred  and  one  millions,  three, 
thousand  and  six. 

33  Write  four  hundred  and  four  septillions,  nine  hundred 
and  three  sextillions,  two  hundred  and  one  quintiliionij,  forty 
i|uadrillious.  tliret*  hundred  and  four 


2'Z 


ADDITIUN. 


ADDITION. 


30.  1.  JcHN  has  two  apples  and  Charles  has  three  ;  how 
many  have  both  ? 

Analysis. — if  John's  apples  be  placed  with  Charles's,  there  will 
1)6  five  apples. 

The  operation  of  finding  how  many  apples  both  have  is  called 
Addition. 

ADDITION  TABLE. 


2  and    0  are 

2  1  3  and    0  are 

3 

4  and    0  are 

4 

5  and 

0  are    5 

2  and     I  are 

3 

3  and     1  are 

4 

4  and     i  are 

5 

5  and 

1  are    6 

2  and    2  are 

4 

3  and    2  are 

5 

4  and    2  are 

6 

5  and 

2  are    7 

2  and    3  are 

5 

3  and    3  are 

6 

4  and    3  are 

7 

5  and 

3  are    8 

2  and    4  are 

6 

3  and    4  are 

7 

4  and    4  are 

8 

5  and 

4  are    9 

2  and     5  are 

7 

3  and    5  are 

8 

4  and    5  are 

9 

Sand 

5  are  lO 

2  and    fi  are 

8 

3  and     6  are 

9 

4  and    6  are 

10 

5  and 

6  are  11 

2  and     7  are 

9 

3  and     7  are 

10 

4  and    7  are 

11 

5  and 

7  are  12 

2  and     8  are 

10 

3  and    8  are 

11 

4  and    8  are 

12 

5  and 

8  are  13 

2  and    9  are 

11 

3  and    9  are 

12 

4  and    9  are 

13 

5  and 

9  are  14 

2  and  10  are 

12 

3  and  10  are 

13 

4  and  10  are 

14 

5  and 

10  are  15 

6  and    0  are 

6 

7  and    0  are 

7 

8  and    0  are 

8 

9  and 

0  are    9 

6  and     1  are 

7 

7  and     1  are 

8 

8  and     1  are 

9 

9  and 

1  are  10 

6  and    2  are 

8 

7  and    2  are 

9 

8  antl    2  are 

10 

9  and 

2  are  1 1 

6  and    3  are 

9 

7  and    3  are 

10 

8  and    3  are 

11 

9  and 

3  are  12 

6  and    4  are 

10 

7  and    4  are 

11 

8  and    4  are 

12 

9  and 

4  are  13 

6  and    5  are 

11 

7  and    5  are 

12 

8  and    5  are 

13 

9  and 

5  are  14 

6  and    6  are 

12 

7  and    6  are 

13 

8  and    6  are 

14 

9  and 

6  are  15 

6  and     7  are 

13 

7  and    7  are 

14 

8  and    7  are 

15 

9  and 

7  are  16 

6  and    8  are 

14 

7  and    8  are 

15 

8  and    8  are 

16 

9  and 

8  are  17 

6  and    9  are 

15 

7  and    9  are 

16 

8  and    9  are 

17 

9  and 

9  are  18 

6  and  10  are 

16 

7  and  10  are 

17 

8  and  10  are 

18 

9  and  10  are  19 

2. 

both 

3 


James  has  5  marbles  and  William  7  :  how  many  have 


Mary  has  6  pins  and  Jane  9 :  how  many  have  both  ? 
How  many  are  4  and  5  and  3  ? 
How  many  are  6  and  4  and  9  '^ 


2  and  8  ]  5  and  5 


4. 
5. 

6.  How  miany  are  3  and  7  ?  4  and  6 
and  n   10  and  0?  0  and  10  ? 

7.  How  many  are  G  and  3  and  9  ]     How  many  are  lb  and 
'{   \^  and  3'Mt  and  5? 


SLMPLJi;    NUMJilOiS.  23 

8.  James  had  9  cents  and  Henry  gave  him  eight  more  : 
how  many  had  he  in  all  ? 

PKINCIPLES    AND    EXAMPLES. 

31.  James  has  3  apples  and  John  4  :  how  many  have 
both  1     Seven  is  called  the  sum  of  the  numbers  3  and  4 

The  SUM  of  two  or  more  numbers  is  a  number  which  conr 
tains  as  many  units  as  all  the  numbers  taken  together. 

Addition  is  the  operation  of  finding  the  sum  of  two  or  more 
numbers. 

OP    THE    SIGNS. 

32.  The  sign  -|-  is  called  plus,  which  signifies  more. 
When  placed  between  two  numbers  it  denotes  that  they  are 
to  be  added  together. 

The  sign  rz:  is  called  the  sign  of  equality.  When  placed 
between  two  numbers  it  denotes  that  they  are  equal ; 
that  is,  that  they  contain  the  same  number  of  units.  Thus  : 
3  +  2  =  5. 

2  +  3=     how  many  ? 

1+2  +  4=     how  many? 

2  +  3+5+1=     how  many? 

6  +  7  +  2+3=     how  many? 

1  +  6  +  7  +  2  +  3=     how  many? 

l-(_2  +  3  +  4  +  5  +  6  +  7  +  8  +  9=      how  many? 

1.  James  has  14  cents,  and  John  gives  him  21  :  how  many 
will  he  then  have  ? 

OPERATION 

14 
Analysis. — Having  written  the  numbers,  as  at  the       21 
right  of  the  page,  draw  a  line  beneath  them.  gg  cents 

The  first  number  contains  four  units  and  1  ten,  the  second  ] 
unit  and  two  tens.  We  write  the  units  in  ore  column  and  the 
tens  in  the  column  of  tens. 


31.  What  is  the  sum  of  two  or  more  numbers  ?     What  is  addition  ^ 

32.  What  is  the  sign  of  addition  ^  What  is  it  called  !  \A'hat  doe* 
it  signify  \  Express  the  srtgn  of  equality  1  When  placed  between  two 
numbers  what  does  it  show  1  When  is  a  number  equal  to  Uic  sutii 
of  other  iiuinbers  :      'iive  an  example  1 


24  ADDITION. 

We  then  begin  at  the  right  hand,  and  say  1  and  4  are  .5,  whicli 
we  set  down  below  the  line  in  the  units'  place.  Wc  then  add 
the  tens,  and  write  the  sum  in  the  tens'  place.  Hence,  the  sum 
is  3  tens  and  5  units,  or  35  cents. 

OPERATION. 

24 

2.  John  has  24  cents,  and  William  62  :  how  62 
many  have  both  of  them?  86 

OPERATION. 
1  fiO 

3.  A  farmer  has  160  sheep  in  one  field,  20  in  ^n 
another,  and  16  in  another:  how  many  has  he  .^ 
in  all  ?  _ir 

196 

OPERATION 
328 

171 
491) 


4.   What  is  the  sum  of  328  and  171  ? 


(5.)  (6.)  (7.)  (6.) 

427  329  3034  8094 

242  260  6525  1602 

330  100  236  103 

999 

9.  What  is  the  sura  of  304  and  273  ? 

10.  What  is  the  sum  of  3607  and  4082? 

11.  What  is  the  sum  of  30704  and   471912  ? 

12.  What  is  the  sum  of  398463  and  401536  ? 

13.  If  a  top  costs  6  cents,  a  knife  25  cents,  a  slate  12 
cents  :  what  does  the  whole  amount  to  1 

14.  John  gave  30  cents  for  a  bunch  of  quills,  18  cents  foj 
an  inkstand,  25  cents  for  a  quire  of  paper  :  what  did  the 
whole  cost  him  ? 

15.  If  2  cows  cost  143  dollars,  5  horses  621  dollars,  and  2 
yoke  of  oxen  124  dollars  :  what  will  be  the  cost  of  them  all  ? 

16.  Add  5  units,  6  tens,  and  7  hundreds. 
Analysis. — We  set  down  the  5  units  in  the  place       -S 

of  units,  the  6  tens  in  the  place  of  tens,  and  the   7       ^    .  w 

hciidreds  in  the  place  of  hundreds.     We  then  add  .up,        g  g  - 

and  £ud  the  sum  to  be  765.  ^  S  ^ 

We  must  observe,   that  in  all  cases,  units  of  the       „    ti  "^ 

uwie  order  are  ivritten  in  the  mwut  column.  ' 

7  6  6 


DECIMAL    FUACriONS. 


RULE    FOR    WRITING    DECIMALS. 


185 


Write  the  decimal  as  if  it  were  a  whole  number,  jrrefiz- 
^ng  as  many  ciphers  as  are  necessary  to  make  ih  of  tlt£ 
required  denontirtation. 

RULE    FOR    READING    DECIMALS. 

Read  the  decimal  as  though  it  luere  a  whole  number, 
adding  the  denomination  indicated  by  the  lowest  deciinol 
unit, 

EXAMPLES. 

Write  the  following  numbers  decimally  : 
(1.)  (2.)  (3.)  (4.)  (5.) 

3  16  17  32  165 

Too'       looo'       loooo'       too'       loobo' 

(6.)  (7)  (8)  (9)  (10.) 

IStI^.       y^Sw       IGrHii-       9?gi       ii¥^- 

Write  the  following  numbers  in  figures,  and  then  tiumeratc 
them. 

1.  Forty-one,  and  three-tenths. 

2.  Sixteen,  and  three  millionths. 

3.  Five,  and  nine  hundredths. 

4.  Sixty-five,  and  fifteen  thousandths, 

5.  Eighty,  and  three  millionths. 

6.  Two,  and  three  hundred  millionths 

7.  Four  hundred,  and  ninety -two  thousandths. 

8.  Three  thousand,  and  twenty-one  ten  thousandths, 

9.  Forty-seven,  and  twenty-one  hundred  thousandths, 

10.  Fifteen  hundred,  and  three  millionths. 

1 1 .  Thirty-nine,  and  six  hundred  and  forty  thousandths, 

12.  Three  thousand,  eight  hundred  and  forty  millionths, 

13.  Six  hundred  and  fifty  thousandths. 

803.  Does  the  value  of  the  unit  of  a  figure  depend  upon  the  place 
which  it  occupies  ]  How  does  the  value  change  from  the  left  towards 
the  right  !  What  do  ten  units  of  any  one  place  make  '  How  do  the 
units  of  the  places  increase  from  the  right  towards  the  left  \  How  may 
whole  numbers  be  joined  with  decimals  1  What  is  such  a  number 
called  \  Give  the  rule  for  writing  decimal  fractions.  Give  the  rule 
for  reading  decimal  lracliou&. 

7 


186  UNTTED    STATKB    MONEY. 

UNITED  STATES  MONEY. 

204.  The  denominations  of  United  States  Money  correspond 
to  the  decimal  division,  if  we  regard  1  dollar  as  the  unit.        4 

For,  the  dimes  are  tenths  of  the  dollar^  ike  cents  are  hun- 
dredths of  the  dollar^  and  the  mills^  being  tenths  of  the  cenl, 
are  thousandths  of  the  dollar. 

EXAMPLES. 

1.  Express  $39  and  39  cents  and  7  mills,  decimally. 

2.  Express  $12  and  3  mills,  decimally. 

3.  Express  |147  and  4  cents,  decimally. 

4.  Express  $148  4  mills,  decimally. 

5.  Express  $4  6  mills,  decimally. 

6.  Express  |9  6  cents  9  mills,  decimally. 

7.  Express  $10  13  cents  2  mills,  decimally. 

ANNEXING  AND  PREFIXING  CIPHERS. 

205.  Annexing  a  cipher  is  placing  it  on  the  right  of  a 
number. 

If  a  cipher  is  annexed  to  a  decimal  it  makes  one  tnore  deci- 
mal place,  and  therefore,  a  cipher  must  also  be  added  to  the 
denomijiator  (Art.  202). 

The  numerator  and  denominator  will  therefore  have  been 
multiplied  by  the  same  number,  and  consequently  the  value 
of  the  fraction  will  not  be  changed  (Art.  161) :  hence, 

Annexing  ciphers  to  a  decimal  fraction  does  not  alter  ita 
value. 

We  may  take  as  an  example,  .3=i  j^. 

If  we  annex  a  cipher  to  the  numerator,  we  must,  at  tho 
same  time,  annex  one  to  the  denominator,  which  gives, 


204.  If  the  denominations  of  Federal  Money  be  expressed  decimally, 
what  is  the  unit  i  What  part  of  a  dollar  is  1  dime  1  What  part  of  a 
dime  is  a  cent  ]  What  part  of  a  cent  is  a  mill  1  What  part  of  a  dollar 
is  1  cent  1     I  mill  1 

205.  When  is  a  cipher  annexed  to  a  number  1  Does  the  annexing 
of  ciphers  to  a  decimal  alter  its  value  1  Why  not  1  What  does  three 
tenths  become  by  annexing  a  cipher  1  W^hat  by  annexing  two  ciphers  \ 
Three  ciphers'?  What  does  8  tenths  become  by  annexing  a  cipher  ^  By 
uiiuexhig  two  ciphers  !     IJy  annexing  three  ciphers  1 


DECIMAT.   F1JA0T10N6.  187 

.3  =     Y^     =  .30       by  annexing  one  cipher, 
,3  =    tVo*o     =  '300     by  annexing  two  ciphers, 
t         .o  =   j-^^   ~  .3000  by  annexing  three  ciphers. 

Also,  .5=T%--r:.50=T^^  =  .500.../^V 

Also,  .8  =  .80=::.800==.8000=:.80000. 

206.  Prefixing  a  cipher  is  placing  it  on  the  left  of  a 
number. 

If  ciphers  are  prefixed  to  the  numerator  of  a  decimal  frac- 
tion, the  same  number  of  ciphers  must  be  annexed  to  the 
denominator.  Now,  the  numerator  will  remain  unchanged 
while  the  denominator  will  be  increased  ten  times  for  every 
cipher  annexed  ;  and  hence,  the  value  of  the  fraction  will  be 
dimi/dshed  ten  times  for  every  cipher  prefixed  to  the  nume- 
rator (Art.  160). 

Prefixing  ciphers  to  a  decimal  f? action  di^ninishes  its 
value  ten  times  Jar  every  cipher  prejixed. 

Take,  for  example,  the  fraction  .2=:^^. 
.2  becomes     j^^q      =  ,02       by  prefixing  one  cipher, 
.2  becomes    yo^     =  .002     by  prefixing  two  ciphers, 
.2  becomes  i^Vin^   =  .0002  by  prefixing  three  ciphers  : 

in  which  the  fraction  is  diminished  ten  times  for  every  cipher 

prefixed. 

ADDITION  OF  DECIMALS. 

207.  It  must  be  remembered,  that  only  units  of  the  same 
kind  can  he  added  together.  Therefore,  in  setting  down 
decimal  r>nmbers  for  addition,  figures  expressing  the  same 
unit  must  be  placed  in  the  same  column. 

206.  When  is  a  cipher  prefixed  to  a  number  1  When  prefixed  to  a 
decimal,  does  it  increase  the  numerator  ?  Does  it  increase  the  denomi- 
nator f  \V  hat  efiiect  then  has  it  on  the  value  of  the  fraction  ?  What 
do  .2  become  by  prefixing  a  cipher  1  By  prefixing  two  ciphers  !  By 
prefixing  three  ?  What  do  .07  become  by  prefixing  a  cipher  !  By  pre- 
fixing two  ?     B}'  prefixing  three  1     By  prefixing  four  1 

207.  What  parts  of  unity  may  be  added  together  1  How  do  yon  set 
down  the  numbers  for  addition  I  How  will  the  decimal  points  fall  I 
How  do  you  then  odd  1  How  many  decimal  places  du  you  pohit  oil  in 
ll;t  tuin  ' 


188  A.DDrrioN  ok 

The  addition  of  decimals  is  then  made  in  the  same  manner 
as  that  of  whole  numbers. 

I.  Find  the  sum  of  37.04,  704.3,  and  .0376. 

OPERATION. 

Place  the  decimal  points  in  the  same  column:  37.04 

this  brings   units  of  the  same   value  in   the   same        704.3 
column  :  then  add  as  in  whole  numbers  :  hence,  .0376 

741.3776 
Rule. — I.   Set  down  the  numbers  to  he   added  so  that 
figures  of  the  same  unit  value  shall    stand  in  the  same 
column. 

II.  Add  as  in  simple  nur/ibers,  and  point  off  in  the  sum^ 
from  the  right  hand^  as  many  places  for  decimals  as  are  equal 
to  the  (jreaiest  number  of  places  in  any  of  the  numbers  added. 

Proof. — The  same  as  'n  simple  numbers. 

EXAMPLES. 

1.  Add  4.035,  763.196,  445.3741,  and  91.3754  together. 

2.  Add  365.103113,  .76012,  1.34976,  .3549,  and  61.11 
together. 

3.  67.407  +  97.0044-4-1- .6  +  . 06-1- .3 

4.  .0007-f-1.0436-|-.4-h.05  +  .047 

5.  .0049  +  47.0426  +  37.0410  +  360.0039  =  444.0924. 

6.  What  is  the  sum  of  27,  14,  49,  126,  999,  .469,  and 
.2614] 

7.  Add  15,  100,  67,  1,  5,  33,  .467,  and  24.6  together. 

8.  What  is  the  sum  of  99,  99,  31,  .25,  60.102,  .29,  and 
100.347  \ 

9.  Add  together  .7509,  .0074,  69.8408,  and  .6109. 

10.  Required  the  sum  of  twenty-nine  and  3  tenths,  four 
hundred  and  sixty-five,  and  two  hundred  and  twenty-one 
thousandths. 

1 1 .  Required  the  sum  of  two  hundred  dollars  one  dime 
three  cents  and  9  mills,  four  hundred  and  forty  dollars  nine 
mills,  and  one  dollar  one  dime  and  one  mill. 

12.  W^hat  is  the  sum  of  one-tenth,  one  hundredth,  aufl  one 
theusaudth  ? 


UICOTMAL    FKAUTTONS.  W* 

13.  What  is  the  sum  of  4,  and  6  ten-thousandths'? 

14.  Required,  in  dollars  and  decimals,  the  sum  of  one  dollar 
one  dime  one  cent  one  mill,  six  dollars  three  mills,  four  dol- 
lars eight  cents,  nine  dollars  six  mills,  one  hundred  dollars  six 
dimes,  nine  dimes  one  mill,  and  eight  dollars  six  cents. 

15.  What  is  the  sum  of  4  dollars  6  cents,  9  dollars  3  mills, 
14  dollars  3  dimes  9  cents  1  mill,  104  dollars  9  dimes  9  cents 
9  mills,  999  dollars  9  dimes  1  mill,  4  mills,  6  mills,  and  1 
mill? 

16.  If  you  sell  one  piece  of  cloth  for  |i4,25,  another  for 
$5,075,   and  another  for  $7,0025,  how  much  do   you  get  for 

ain 

17.  What  is  the  amount  of  $151,7,  $70,602,  $4,06,  and 
$807,26591 

18.  A  man  received  at  one  time  $13,25  ;  at  another  $8,4  ; 
at  another  $23,051  ;  at  another  $6  ;  and  at  another  $0,75  : 
how  much  did  he  receive  in  all  ? 

19.  Find  the  sum  of  twenty-five  hundredths,  three  hundred 
and  sixty-five  thousandths,  six  tenths,  and  nine  millionths. 

20.  What  is  the  sum  of  twenty- three  millions  and  ten,  one 
thousand,  four  hundred  thousandths,  twenty-seven,  nineteen 
millionths,  seven  and  five  tenths  1 

21.  What  is  the  sum  of  six  millionths,  four  ten-thousandths, 
19  hundred  thousandths,  sixteen  hundredths,  and  four  tenths'? 

22.  If  a  piece  of  cloth  cost  four  dollars  and  six  mills,  eight 
pounds  of  coffee  twenty-six  cents,  and  a  piece  of  muslin  three 
dollars  seven  dimes  and  twelve  mills,  what  will  be  the  cost 
of  them  all  ? 

23.  If  a  yoke  of  oxen  cost  one  hundred  dollars  nine  dimes 
and  nine  mills,  a  pair  of  horses  two  hundred  and  fifty  dollars 
five  dimes  and  fifteen  mills,  and  a  sleigh  sixty-five  dollars 
eleven  dimes  and  thirty-nine  mills,  what  will  be  their  entire 

coat  ? 

24.  Find  the  sum  of  the  following  numbers  :  Sixty-nine 
thousand  and  sixty-nine  thousandths,  forty-seven  hundred  and 
forty-seven  thousandths,  eighty-five  and  eighty-five  hun- 
dredths, six  hundred  and  forty-nine  and  six  hundred  aii'V 
ibrty-iiine  leii-tliousandtlis  ? 


190  BUBTli ACTION    OP 


SUBT-KACTION  OF  DECIMALS. 

20b    Subtraction  of  Decimal  Fractions  is  the  operation  of 
finding  the  ditlerence  between  two  decimal  numbers. 

I.  From  3.275  to  take  .0879. 

Note.  —  Iii  this   example  a  cipher    is    annexed  operation 

to  the  minuend   to   make  the  number  of  decimal  3.2750 

places  equal  to  the  number  in  the  subtrahend.  This  .0879 

does  not  alter  the  value  of  the  minuend  (Art.  205)  :  "^ilE^^T 

hence,  ^'^^^^ 

Rule. — 1.    Write  the  less  mi?nber  under  the  greater^  so  thai 
Jig u res  of  the  same  unit  value  shall  fall  in  the  same  column. 

II.  Subtract  as  in  simple  numbers^  and  point  off  the  deci- 
mal places  in  the  remainder,  as  in  addition. 

Proof. — Same  as  in  simple  numbers. 

EXAMPLES. 

1.  From  3295  take   0879. 

2.  From  291.10001  take  41.375. 

3.  From  10.000001  take  .1111 IL 

4.  From  396  take  8  ten-thousandth8. 

5.  From  1  take  one  thousandth. 

6.  From  6378  take  one-tenth. 

7.  From  365.0075  take  3  millionths. 

8.  From  21.004  take  97  ten-thousandths. 

9.  From  260.4709  take  47  ten-millionths. 

10.  From  10.0302  take  19  millionths. 

11.  From  2.01  take  6  ten-thousandths. 

12.  From  thirty- five  thousands  take  thirty-five  thousandths. 

15.  From  4262.0246  take  23.41653. 

14.  From  346.523120  take  219.691245943. 

16.  From  64.075  take  .195326. 

lb.  What  is  the  difierence  between  107  and  .0007  1 

17.  What  is  the  difierence  between  i.^  and  .3735  ? 

18.  From  96.71  take  96.709. 


208.  What  is  subtraction  of  decimal  fractions'?  How  do  you  set  down 
the  numbers  for  subtraction  1  How  do  you  then  subtriict  !  How  jnaiiy 
Jeciuiul  p!a«;t!s  do  yo\.x  p<»liit  oHiu  the  reni:.imltr^ 


DECIMAL    VI 


MULTIPLIGA.T10N  OF  DECK 
209.   To  multiply  one  decimal  by  another. 
I.  Multiply  3.05  by  4.102. 

OPERAnoN. 

Analysis. — If  we  change  both  factors  to  vul-  s^-VOS.  — 3.05 

gar  fraciioiis,  the  product  of  the  numerator  will  4  i  o  2  _  ,i  iaq 

be  the  same  as  that  of  the  decimal  numbers,  and  Tooo  —  _: 

the  number  of  decimal  places  will  be  equal  to  the  610 

number   of    ciphers   in   the   two   denominators :  305 

hence,  12  20 

12.51110 

Rule. — Maltiphj  as  in  simple  numbers^  and  point  off  in 
the  product^  from  the  right  hand^  as  many  figures  for  decimals 
US  there  are  decimal  places  in  both  factors;  and  if  there  be 
not  so  many  in  the  product,  siqrply  the  deficiency  by  prefixing 
ciphers. 

EXAMPLES. 

1.  Multiply  3.049  by  .012. 

2.  Multiply  365.491  by  .001. 

3.  Multiply  496.0135  by  1.49.6. 

4.  Multiply  one  and  one  milliontb  by  one  thousandth. 

5.  Multiply' one  hundred  and  forty-seven  millionths  by  one 
millionth, 

6.  Multiply  three  hundred,  and  twenty-seven  hundredths 
by  31. 

7.  Multiply  31.00467  by  10.03962. 

8.  What  is  the  product  of  five-tenths  by  five-tenths  ? 

9.  What  is  the  product  of  five-tenths  by  five-thousandths'? 

10.  Multiply  596.04  by  0.00004. 

11.  Multiply  38049.079  by  0.00008. 

12  What  will  6.29  weeks'  b-^ard  come  to  at  2,75  dollars 
per  week  ? 

13.  What  will  61  pounds  of  sugar  come  to  at   $0,234  per 


pound 


209.  After  multiplying,  how  many  decimal  places  will  you  point  off 
in  the  product  \  When  there  are  not  so  many  in  the  product,  what  de 
ytto  do"?     Give  the  rule  for  thr-  multiplication  of  deci turds. 


192 


UONTKACTlOi^b. 


14.  If  12.836  dollars  are  paid  for  one  barrel  of  flour,  what 
»vill  .354  barrels  cost  1 

15.  What  are  the  contents  of  a  board,  .06  feet  long  and  .06 
wide  ? 

16.  Multiply  49000  by  .0049. 

17.  Bought  1234  oranges  for  4.6  cents  apiece  :  how  much 
iid  they  cost? 

18.  What  will  375.6  pounds  of  coffee  cost  at  .125  dollars 
per  pound  ? 

19.  If  I  buy  36.251  pounds  of  indigo  at  $0,029  per  pound, 
what  will  it  come  to  ? 

20.  Multiply  $89.3421001  by  .00000^8. 

21.  Multiply  1341.45  by  .007. 

22.  What  arethecontentsof  alot  whichis  .004  miles  long 
and  .004  miles  wide? 

23.  Multiply  .007853  by  .035. 

24.  What  is  the  product  of  $26.000375  multiplied  by* 
.00007  ? 

CONTRACTIONS. 

210.  W^hen  a  decimal  number  is  to  be  multiplied  by  10, 
100,  lOOOj  &c.,  the  multiplication  may  be  made  by  removing 
the  decimal  point  as  many  places  to  the  right  hand  as  there 
are  ciphers  in  the  multiplier,  and  if  there  be  not  so  many 
figures  on  the  right  of  the  decimal  point,  supply  the  deficieiicy 
by  annexing  ciphers. 


Thus,  6.79  multiplied  by 


Also,  370.036  multiplied  by 


'10 

1 

67.9 

100 

679. 

-  1000 

'  =  ^ 

6790. 

10000 

67900. 

100000^ 

679000. 

10    1 

3700.36 

100 

37003.6 

1000 

►  =:  • 

370036. 

10000 

3700360. 

100000  I 

3 

7003600. 

210.  How  do  you  multiply  a  decimal  number  by  10,  100,  1000,  Ac. ! 
If  there  are  not  as  many  decimal  figures  as  there  are  ciphers  in  the 
uiultiplier,  what  do  you  do  ^ 


DECIMAL    FRACTIONS.  103 


DIVISION  OF  DECIMAL  FRACTIONS. 

211.  Division  of  Decimal  Fractions  is  similar  to  that  of 
limple  numbers. 

1.   Let  it  be  reqmred  to  divide  1.38483  by  60.21. 

Analysis. — The  dividend  must  be  equal  opkration. 

the  product  of  the  divisor  and  quotient,  60.21)1.38483(23 

(Art.    61);  and    hence  must   contain   as  1.2042 

many  decimal  places   as    both  of  them  ;  1  80r*V 
therefore, 

There  must  he  as  many  decimal  places  in  

the  quotient  as  the  decimal  places  in  the  divi-  Ans.  .023 
dend  exceed  those  in  the  divisor  :  hence, 

E.ULE. — Divide  as  in  simple  nnmbers,  and  point  off  in  the 
quotient^  from  the  right  hand,  as  many  places  for  dechnals  as 
the  decimal  J) taces  i?i  the  dividend  exceed  those  in  the  divisor  ; 
and  if  there  are  not  so  many,  supply  the  deficiency  by  prefiic- 
ing  ciphers, 

EXAMPLES. 


1.  Divide  2.3421  by  2.11. 

2.  Divide  12.8256f  by  3.01. 

3.  Divide  33.66431  by  1.01. 


4.  Divide  .010001  by  .01. 

5.  Divide  8.2470  by  .002. 

6.  Divide  94.0056  by  .08. 


7.  What  is  the  quotient  of  37.57602,  divided  by  3  ;  by  .3  , 
by  .03  ;  by  .003  ;  by  .0003  ? 

8.  What  is  the  quotient  of  129  75896,  divided  by  8  ;  by 
.08  ;  by  .008  ;  by  .0008  ;  by  .00008  X 

9.  What  is  the  quotient  of  187.29900,  divided  by  9 ;  by 
.9  ;  by  .09  ;  by  .009  ;   by  .0009  ;  by  .00009  ? 

10.  What  is  the  quotient  of  764.2043244,  divided  by  6  ; 
by  .06  ;  by  .006  ;  by  .0006  ;  by  .00006  ;  by  .000006? 

Note. — 1.  When  there  are  more  decimal  places  in  the  divisor 
than  in  the  dividend,  annex  ciphers  to  the  dividend  and  make  the 
decimal  places  equal ;  all  the  figures  of  the  quotient  will  then  bs 
whole  numbers. 

211.  How  does  the  number  of  decimal  places  in  the  dividend  c(im- 
pare  with  thai  in  the  divi.sor  and  (juotient  !  How  do  you  determine 
the  nural)er  of  decimal  places  in  the  quotient "!  •  If  the  divisor  contains 
four  places  and  the  vlividend  six,  how  many  in  the  quotient  !  If  the 
divisor  contains  three  places  and  the  dividend  five,  liow  many  in  the 
quotient  ?  Give  the  rule  for  the  division  of  decimals. 
13 


lU 


DIVISION    OF 


EXAMPLES. 


1    Divide  4397.-1  by  3.49. 


Note. — We  annex  one  0  to 
trie  dividend.  Had  it  contained 
no    decimal    place    we    should 


have  annexed  two. 


OP  K  RATION. 

3.49)4397.40(1260 
34  9 

~907 
698 
2094 
2094 


Ans.  1260. 


2.  Divide  2194.02194  by  .100001. 

3.  Divide  9811.0047  by  .325947. 

4.  Divide  .1  by  .0001.  |       5.  Divide  10  by  .15. 
6.  Divide  6  by  .6  ;  by  .06  ;  by   .006  ;  by  .2  ;  by   3 


by 


by  .003;  by  .5;  by  .05;  by  .005. 

Note. — 2.  When  it  is  necessary  to  continue  the  division  farther 
than  the  figures  of  the  dividend  will  allow,  we  annex  ciphers,  and 
consider  them  as  decimal  places  of  the  dividend. 

When  the  division  docs  not  terminate,  we  annex  the  plus  sign 
to  show  that  it  may  be  continued  ;  thus  .2  divided  by  .3  =  .666-f. 


EXAMPLES. 


1.  Diwde  4.25  by  1.25. 

Anaia'sts. — In  this  example  we  annex  one  0, 
and  then  the  decimal  places  m  the  dividend  will 
exceed  those  in  the  divisor  by  1. 


OPERATION. 

1.25)4.25(3.4 
3.75 


500 
500 

Ans.  3.4. 


4    Divide  580.4  by  375. 

5.  Divide  94.0369  by  81.032. 


2.  Divide  .2  by  .6. 

3.  Divide  37.4  by  4.5. 

Note. — 3.  When  any  decimal  number  is  to  be  divided  by  10, 
100.  1000,  &c.,  the  division  is  made  by  removing  the  decimal 
point  as  tmwy  places  to  the  left  as  there  are  O'.s  in  the  divisor  ;  and 
if  there  be  not  hO  many  tigures  on  the  left  of  the  decimal  point 
the  deficiency  is  supplied  by  prefixing  ciphers. 


27.69  divided  bv 


10    ] 

r  2.769 

100    1 

.2769 

1000  f-' 

.02769 

10000  J 

.002769 

642.89  divided  by 


ILC 

IMAL    KKACTIONS. 

fio       1 

r  64.289 

100 

6.4289 

1000 

>  =  < 

.64289 

10000 

.064289 

100000 

.0064289 

196 


QUESTIONS    IN    THB    PRECEDING    RULES. 

1     If  I  divide  .6  dollars   among  94  men,  how  much  will 
eaoh  leceive  ? 

2.    1  gave  28  dollars  to  267  persons  :  how  much  apiece  ? 
Divide  6.35  by  .425. 
What  is  the  quotient  of  136.2678  divided  by  2.25  1 


Divide  a  dollar  into  12  equal  parts. 


Divide  .25  ol'3.26  into  .034  of  3.04  equal  parts. 
How  many  times  will  .35  of  35   be  contained   in  .024 
01241 

8.  At  .75  dollars  a  bushel,  how  many  bushels  of  rye  can 
be  bought  for  141  dollars'? 

9.  Bought  12  and  15  thousandths  bushels  of  potatoes  for 
33  hundredths  dollars  a  bushel,  and  paid  in  oats  at  22  hun- 
dredths of  a  dollar  a  bushel :  how  many  bushels  of  oats  did  it 
take? 

10.  Bought  53.1  yards  of  cloth  for  42  dollars  :  how  much 
was  it  a  yard  ? 

11.  Divide  125  by  .1045. 

12.  Divide  one  millionth  by  one  billionth. 

13.  A  merchant  sold  4  parcels  of  cloth,  the  first  contained 
127  and  3  thousandths  yards  ;  the  2d,  6  and  3  tenths  yards  ; 
the  3d,  4  and  one  hundredth  yards  ;  the  4th,  90  and  one 
millionth  yards  :  how  many  yards  did  he  sell  in  all  ? 

14.  A  merchant  buys  three  chests  of  tea,  the  first  contains 
60  and  one  thousandth  pounds  ;  the  second,  39  and  one  ten 
thousandth  pounds  ;  the  third,  26  and  one  tenth  pounds  :  how 
much  did  he  buy  in  all  ? 

Note. —  1.  If  there  are  more  decimal  places  in  the  divisor  than  in  the 
divitlend,  what  do  you  do  !  M'hat  will  the  figures  of  the  quotient  then 
be  ! 

2.  How  do  you  continue  the  division  after  you  have  hrout'^ht  down  all 
the  llgures  of  the  dividend  I  What  sign  do  you  place  after  the  quo- 
tient !      What  does  it  show  1  / 

3.  How  do  you  divide  a  decimal  fraction  by  10,  100,  1000,  &c. ! 


196  DIVISION  Ot- 

is. What  is  the  sum  of  ^20  and  three  hundredths  ;  S4 
and  one-tenth,  $6  and  one  thousandth,  and  $18  and  one 
hundredth  ? 

16.  A  puts  in  trade  $504,342  ;  B  puts  in  $350.1965  ;  C 
puts  in  $100.11;  D  puts  in  899.334;  and  E  puts  in 
19001.32  :  what  is  the  whole  amount  put  in  1 

17.  B  has  $936,  and  A  has  $1,3  dimes  and  1  mill  :  how 
much  more  money  has  B  than  A  1 

18.  A  merchant  buys  37.5  yards  of  cloth,  at  one  dollar 
twenty-five  cents  per  yard  :  how  much  does  the  whole 
come  to  ? 

19.  If  12  men  had  each  $339  one  dime  9  cents  and  3 
mills,  what  would  be  the  total  amount  of  their  money  ? 

20.  A  farmer  sells  to  a  merchant  13.12  cords  of  wood  at 
$4,25  per  cord,  and  13  bushels  of  wheat  at  $1,06  per  bushel  : 
he  is  to  t^ke  in  payment  13  yards  of  broadcloth  at  $4,07  per 
yard,  and  the  remainder  in  cash  :  how  much  money  did  he 
receive  1 

21.  If  one  man  can  remove  5.91  cubic  yards  oi"  earth  in  a 
day,  how  much  could  nineteen  men  remove  ? 

22.  What  is  the  cost  of  8.3  yards  of  cloth  at  $5,47  per 
yard  ? 

23.  If  a  man  earns  one  dollar  and  one  mill  per  day,  how 
much  will  he  earn  in  a  year  of  313  working  days  ? 

24.  What  will  be  the  cost  of  375  thousandths  of  a  cord  of 
wood,  at  $2  per  cord  ? 

25.  A  man  leaves  an  estate  of  $1473.194  to  be  equally 
divided  among  12  heirs  :  what  is  each  one's  portion  ? 

26.  If  flour  is  $9,25  a  barrel,  how  many  barrels  can  I  buy 
for  $1637,25  ? 

27.  Bought  26  yards  of  cloth  at  $4,37^  a  yard,  and  paid 
for  it  in  flour  at  $7,25  a  barrel  :  how  much  flour  will  pay 
for  the  cloth  1 

28.  How  much  molasses  at  22^  cents  a  gallon  must  be 
given  for  46  bushels  of  oats  at  45  cents  a  bushel  ? 

29.  How  many  days  work  at  $1,25  a  day  must  be  givea 
for  6  cords  of  wood,  worth  $4,12J  a  cord  ? 

30.  What  will  36.48  yards  of  cloth  cost,  if  14.25  yard 
cost  $21,375? 

31.  If  you  can  buy  13.25Z6.  of  cofl^ee  for  $2,50,  liow  much 
can  you  buy  for  $325,50  'i 


DECIMAL    FKAUTIONS.  197 

212.    To  change  a  common  to  a  decimal  fraction. 

The  value  of  a  fraction  is  the  quotient  of  the  numerator, 
divided  by  the  denominator  (Art.  148). 

1.  Reduce  f  to  a  decimal. 

If  we  place  a  decimal  point  alter  the  5,  and  then      operation. 
write  any  number  of  O's,  after  it,  the  value  of  the        8)5.000 
numerator  will  not  be  changed  (Art.  205).  g25 

If,  then,  we  divide  by  the  denominator,  the  quo- 
tient will  be  the  decimal  number :  hence, 

Rule. — Annex  decimal  ciphers  to  the  numerator,  and 
then  divide  by  the  denomitmtor,  pointing  off  as  in  division 
of  decimals. 

EXAMPLES. 

1.  Reduce  flf  to  its  equivalent  decimal. 

OPKRATION. 

125)635(5.08 
We  here  use  two  ciphers,  and  therefore  point  (325 

off  two  decimal  places  in  the  quotient. 


Reduce  the  following  fractions  to  decimals  : 

1.  Reduce  f  to  a  decimal. 

2.  Reduce  ^-^  to  a  decimal. 


3.  Reduce  ^  to  a  decimal. 

4.  Reduce  \  and  x\2Q' 

5.  Reduce^V  li'^"^ToVo• 
6.  Reduce  ^  and  yyg  5-. 

7  RpflllPP    3  14  9  5  7  12  3 

ft  T?pr1npp    8      137.5      3265 

o.  iteauce  g,  g^^^-,  4T21- 

9.  Reduce  ^  to  a  decimal. 


1000 
1000 


10.  Reduce  ^  to  a  decimal. 

11.  Reduce  ^^. 

12.  Reduce  ^-q. 

13.  Reduce  j^}^. 

14.  Reduce  y^i,j. 

15.  Reduce  y^j^. 

16.  Reduce  ^V^- 

17.  Reduce  y^w^J- 

18.  Reduce  f^^^. 


213.   A  decimal  fraction  may  be  changed  to  the  form  of  a 
vulgar  fraction  by  simply  writing  its  denominatoi  (Art.  202). 


212.   How  do  you  change  a  vulgar  to  a  decimal  fraction  ( 

21il.  How  do  you  change  a  decimal  to  the  form  of  a  vulgar  fraction  ? 


198  DE&'UMINAilC   DKOIMALS. 


EXAMPLES. 

1.  What  vulgar  fraction  is  equal  to  .04  ? 

2.  What  vulgar  fraction  is  equal  to  3.067  ? 

3.  What  vulgar  fraction  is  equal  to  8.275? 

4.  What  vulgar  fraction  is  equal  to  .00049  1 

DENOMINATE  DECIMALS 

214.  A  denominate  decimal  is  one  in  which  the  unit  of  the 
fraction  is  a  denominate  number.  Thus,  .5  of  a  pound,  .6  of  a 
shilling,  .7  of  a  yard,  <kc..,  are  denominate  decimals,  in  which 
the  units  are  1  pound,  1  shilling,  1  yard. 

CASE   I. 

215.  To  change  a  denominate  number  to  a  denominate 
decimal. 

1.   Change  9fZ.  to  the  decimal  of  a  £. 

Analysis. — The  denominate  unit  of  the  frac-  operation. 

lion  is  1£=240£/.     Then  divide  'dd.  by  240:  240<i.  =  i;i 

the  quotient.  .0375  of  a  pound  is  the  value  of  240)9(.0375 

9f/.  in  the  decimal  of  a  £  :  hence,  ^^^     £  0375 

Rule. — Reduce  the  unit  of  the  required  fruction  to  the  unit 
of  the  given  denominate  number^  and  then  divide  the  denoini- 
nate  number  by  the  result.^  and  the  quotient  will  be  the  decanal, 

EXAMPLES. 

1.  Reduce  7  drams  to  the  decimal  of  a  lb.  avoirdupois. 

2.  Reduce  26c/.  to  the  decimal  of  a  £>. 

3.  Reduce  .056  poles  to  the  decimal  of  an  acre. 

4.  Reduce  14  minutes  to  the  decimal  of  a  day. 

5.  Reduce  21  pints  to  the  decimal  of  a  peck.. 

6.  Reduce  3  hours  to  the  decimal  of  a  day. 

7.  Reduce  375678  feet  to  the  decimal  of  a  mile. 

8.  Reduce  36  yards  to  the  decimal  of  a  rod. 

9.  Reduce  .5  quarts  to  the  decimal  of  a  barrel. 

10.  Reduce  .7  of  an  ounce,  avoirdupois,  to  the  decimal  of  a 
hundred. 

214.  What  is  a  denominate  decima. ' 

215.  How  do  you  change  a  denominate  number  to  a  denominate 
docimul  1 


DEWOMINATF    DECIMAI.S.     ,  199 

CASE    II. 

216.  To  find  the  value  of  a  decimal  in  integets  of  a  less 
denomination  I . 

I,  Find  the  value  of  .890625  bushels. 

OPERATION. 

,,,.,.       ,     :,    .      ,  ^    .   /  •        .        .890625 
Analysis. — Multiplying  the  decimal  by  4,  (since  4  . 

pecks  make  a  bushel),  we  have  3.5025  peeks.    Mul-      . 

liphing  the  new  decimal  by  8,  [Ance  Squaris  make  3.562500 

a  peck),  we  have   4.5  quarts.     Then,  multiplying  8 

this  last  decimal  by  2,  (since  2  pints  make  a  quart),  4  500000 

we  have  1  pint :  hence,  '             o 

Ans.  3pk.  4qts.  \pt.      1.000000 

Rule. — I.  Multiply  the  decimal  by  that  number  which 
vrill  reduce  it  to  the  next  less  denomination,  pointing  of  as 
in  mMltiplicati'on  of  decimal  froA^lions. 

II.  Multiply  the  decimal  part  of  the  product  as  befwe  ;  and 
so  continue  to  do  until  the  decimal  is  reduced  to  the  required 
denominations.   The  integers  at  the  left  form  the  ansiver 

EkAMPLES. 

1.  A/VTiat  is  the  value  of  002084/<!».  Troyl 

2.  What  is  the  value  oi'  .625  of  a  cwt.  1 

3.  What  is  the  value  of  .025  of  a  gallon  1 

4.  What  is  the  value  of  £.3375  ? 

5.  What  is  the  value  of  .3375  of  a  ton  ? 

6.  What  is  the  value  of  .05  of  an  acre? 

7.  What  is  the  value  of  .875  pipes  of  wine  ? 

8.  What  is  the  value  of  .125  hogsheads  of  beer? 

9.  What  is  the  value  of  .375  oi  a  year  of  365  days  1 
10.  What  is  the  value  of  .085  of  a  X  ? 

i  1.  What  is  the  value  of  .86  of  a  cwt.  ? 

12.  From  .82  of  a  day  take  .32  of  an  hour. 

13.  What  is  the  value  of  1.089  miles? 

14.  What  is  the  value  of  .09375  of  a  pound,  avoirdupois? 

15.  What  is  the  value  of  .28493  of  a  year  of  365  days  ? 

16.  What  is  the  value  of  £1.046  ? 

17.  What  is  the  value  of  £1.88  ? 

216.  How  do  you  find  flie  value  of  a  decimal  in  integers  of  a  lest 
denomination  \ 


200  ,    DENOMINATE    DECIMALS. 


CASE  m. 

217.   To  reduce  a  compound  denominate  number  to  a 
decimat  or  mixed  nutnher. 

1.  Reduce  £l  45.  ^\d.  to  the  decimal  of  a£. 


OPERATION. 


Analvsis. — Reducing  the  \d.  to  a  decimal 

(Art,  216).  and  annexing  the  result  to  the  9rf.,  sTr™  V^T 

we  have  9  75^i.     Dividing  9.75d.  by  12,  (since  qs  /  ~  q  7^  / 

12  pence=l.v  ),  and  annexing  the  quotient  to  *             ' 

the4s.  we.have  4.81 2o5.  Then,  dividing  by  20  }2\9.'75d 

(since   2''s  —  £1,)  and  annexing  the  quotient  20U  RT^'i*: 

to  the  £J ,  we  have  £1.240625 :  ^ 

Ans.  £1  is.  9f6Z:  =  1.240625£. 

Rule  —Divide  the  Imvest  denomination  by  as  many  units 
as  maki  a.  unit  of  the  next  higher,  and  annex  the  quotient 
as  a  decimal  to  that  higher  :  theft  divide  as  before,  and  so 
co?itimu  to  do,  until  the  decimal  is  reduced  to  the  required 
denomination. 

EXAMPLES. 

1.  Reduce  iwk.  6da.  5hr.  30m.  46s.  to  the  denomination 
of  a  wetfk. 

2.  Reduce  2lb.  5oz.  ]2pivt.  \6gr.,  to  the  denomination  of  a 
pound. 

3.  Re^iuce  3  feet  9  inches  to  the  denomination  of  yards. 

4.  Reduce  lib.  {2dr.,  avoirdupois,  to  the  denomination  of 
pounds. 

5.  Reduce  5  leagues  2  furlongs  to  the  denomination  of 
leagues. 

6.  Reduce  Abu.  Zpk.  4qt.  \pt.  to  the  denomination  of 
bushels. 

7.  Reduce  5oz.  I3pwt.  I2gr.  to  the  decimal  of  a  pound. 

8.  Reduce  Idcwt.  3qr.  2^lb.  to  the  decimal  of  a  ton. 

9.  Reduce  5A.  3R.  2lsq.  rd.  to  the  denomination  of  acres. 

10.  Reduce  11  pounds  to  tlie  decimal  of  a  ton. 

11.  Reduce  3da.  I2^sec.  to  the  decimal  of  a  week. 

12.  Reduce  lAbu.  o'^qt.  to  the  decimal  of  a  chaldron. 

13.  Reduce  Itri.  Ifur.  \r.  to  the  denomination  of  miles. 


217.  How   do   you    reduce   a   compound  denouiiuate  number  to  11 
il«cuiiul  ^ 


ANALYSIS.  201 


ANALYSIS. 

218.  An  analysis  of  a  proposition  is  an  examination  of  its 
separate  parts,  and  their  connections  with  each  other. 

The  sokition  of  a  question,  by  analysis,  consists  in  an  exami- 
nation of  its  elements  and  of  the  relations  which  exist  between 
these  elements.  We  determine  the  elements  and  the  rela- 
tions which  exist  between  them,  in  each  case,  by  examining 
the  nature  of  the  question. 

In  analyzing,  we  reason  from  a  given  number  to  its  unit^ 
and  then  Irom  this  unit  to  the  required  number. 

EXAMPLES. 

1.  If  9  bushels  of  wheat  cost  18  dollars,  what  will  27 
bushels  cost  ? 

Analysis. — One  bushel  of  wheat  will  cost  one  ninth  as  much  as 
9  bushels.  Since  9  but^hels  cost  18  dollars,  1  bushel  will  cost  ^ 
of  18  dollars,  or  2  dollars  ;  27  bushels  will  cost  27  times  as  much 
as  1  bushel:  that  is,  27  times  \  of  18  dollars,  or  54  dollars: 
therefore,  if  9  bushels  of  wheat  cost  18  dollars,  27  bushels  will 
cost  54  dollars. 

OPERATION. 

2 


^xjx^::=$54;     Or, 
10       1 


*^  3 


54  Ans. 


Note. — 1.  We  indicate  the  operations  to  be  performed,  and 
then  cancel  the  equal  factors  (Art.  141). 

219.  Although  the  currency  of  the  United  States  is  ex- 
pressed in  dollars  cents  and  mills,  still  in  most  of  the  States 
the  dollar  (always  valued  at  100  cents),  is  reckoned  in  shil- 
lings and  pence  ;  thus, 

In  the  New  England  States,  in  Indiana,  Illinois,  Missouri,  Vir- 
ginia, Kentucky,  Tennessee,  Missi.«sippi  and  Texas,  the  dollar  is 
reckoned  at  6  shillings  :  In  New  York,  Ohio  and  Michigan,  at  8 
shillings  :  In  New  Jersey,  Pennsylvania,  Delaware  and  Mary- 
land, at  75.  6(f :  In  South  Carolina,  and  Georgia,  at  45.  %d.  :  In 
Canada  and  Nova  Seolia,  at  5  shillings. 

218.  What  is  an  analysis  ]  In  what  does  the  solution  of  a  question 
by  analysis  consist  1  How  do  we  determine  the  elements  and  their 
lelatiuns  \     Huw  du  we  reason  in  analyzing];  1 


202 


ANALYSIS. 


Note. — In  many  of  the  States  the  retail  price  of  articles  is  given 
in  shillings  and  pence,  and  the  result,  or  cost,  required  in  dollars 
and  cents. 

2.  What  will  12  yards  of  cloth  cost,  at  5  shillings  a  yard, 
New  York  currency  ? 

Analysis. — Since  I  yard  cost  5  shillings  12  yards  will  cost  12 
times  5  shillings,  or  60  shillings  :  and  as  8  shillings  make  1  dollar, 
New  York  currency,  there  will  be  as  many  dollars  as  8  is  contain 
ed  times  in  60=$7i. 


OPERATION. 


B. 


6x12-^8  =  17,50;       Or, 


h 

5 

2 

15=  V 

17,50. 


$7,50. 


Note. — The  fractional  part  of  a  dollar  may  always  be  reduced 
to  cents  and  mills  by  annexing  two  or  three  ciphers  to  the  nume- 
rator and  dividing  by  the  denominator  ;  or,  which  is  more  conve- 
nient in  practice,  annex  the  ciphers  to  the  dividend  and  continue 
the  division. 

3.  What  will  be  the  cost  of  56  bushels  of  oats  at  3s.  3d.  a 
bushel,  New  York  currency  ? 


4 

$ 

13 

4 

91 

PKRATION. 

4 

Or.                   4_ 

91 

^22,75  Ans 

22,75. 

Note. — When  the  pence  is  an  aliquot  part  of  a  shilling  the 
price  may  be  reduced  to  an  improper  fraction,  which  will  be  the 
multiplier  :  thus.  3s.  3d.  =  3ks.=  ^s.  Or:  the  shillings  and  pence 
may  be  reduced  to  pence  :  thus,  3s.  3d.  =  39d..  in  which  case  the 
product  will  be  pence,  and  must  be  divided  by  96,  the  number  of 
pence  in  1  dollar  :  hence, 

220.   To  find  the  cost  of  articles  in  dollars  and  cents. 


219.  In  what   is  the  currency  of  the  States  expressed] 
the  currency  of  the  States  often  reckoned  \ 

220.  How  do  you  find  the  cost  of  a  coinmoditv 


In  what  ic 


ANALYSIS. 


Wii 


Multiply  the  commodity  by  the  prire  and  divide  the  product 
by  the  value  of  a  dollar  reduced  to  tlie  same  denominational 
unit. 

4.  What  will  18  yards  of  satinet  cost  at  3s.  9d.   a  yard 
Pennsylvania  currency  ? 

OPERATION. 


$9. 


Or, 


$9  Ans. 


Note. — The  above  rule  will  apply  to  the  currency  in  any  of 
ihe  States.  In  the  last  example  the  multiplier  is  ^s.  9d.  =  3ls 
=  ^5.  or  45d.     The  divisor  is  75.  6d.=:7 hs.=  '^s.=9i)d. 

5.  "What  will  7^/6.  of  tea  cost  at  6s.  8d.  a  pound,  Ne-w 
En^'and  currency  ? 


OPERATION. 


3? 


u 


10 


Or. 


n 


$0 


10 


25 


3  i   25  =  \^=z^b:S33-\- 


$8.333+Jms. 


6.  What  will  be  the  cost  of  \20yds.  of  cotton  cloth  at   Is. 
6d.  a  yard,  Georgia  currency  ? 

7.  What  wilJ  be  the  cost  in  New  York  currency? 

8.  What  will  be  the  cost  in  New  England  currency  ? 

9.  What  will  be  the  cost  of  75   bushels  of  potatoes  at  3s 
6d.,  New  York  currency  ? 

10.  What  will  it  cost  to  build  148   feet  of  wall  at  Is.  8d, 
per  loot,  N.  Y.  currency  1 

11.  What  will  a  load  of  wheat,  containing  46^  bushels, 
come  to  at  lO.v.  8d.  a  bushel,  N.  Y.  currency  ? 

12.  What  will  7  yards  of  Irish  linen  cost  at  3s.  Ad.  a  yard, 
Penn.  currency  ? 

13.  How  many  pounds  of  butter  at  Is.  4d.  a  pound  must 
be  given  for  12  gallons  of  molasses  at  2.s-.  8d.  a  gallon  ? 


204  ANALYSIS. 

OPERATION. 


Or, 


I  24/6. 

Note. — The  same  rule  applies  in  the  last  example  as  in  the 
preceding  ones,  except  that  the  divisor  is  the  price  of  the  article 
received  in  payment,  reduced  to  the  same  unit  as  the  price  of  the 
article  bought.  • 

14.  What  will  be  the  cost  of  \2cwt.  of  sugar  at  9af.  per  lb, 
N.  Y.  currency  ? 


OPERATION. 

25 
9 


Note. —  Reduce  the    cwts.   to    Ihs.   by  ^ 
multiplying  by  4  and  then  by  25,     Then       a, 
multiply    by  the    price  per    pound,   and  ! 

then   divide  by  ihe   value  of  a  dollar  in 
the    required    currency,    reduced    to    the  2~i225 

same  denomination  as  the  price.  

Ms.  $112,50 

15.  What  will  be  the  cost  of  9  hogsheads  of  molasses  at  Is, 
3c?.  per  quart,  N.  E.  currency? 

16.  How  many  days  work  at  Is.  6d.  a  day  must  be  given 
for  12  bushels  of  apples  at  3s. 9d.  a  bushell 

17.  Farmer  A  exchanged  35  bushels  of  barley,  worth  6s. 
4d.,  with  farmer  B  tor  rye  worth  7  shillings  a  bushel  :  how 
many  bushels  of  rye  did  farmer  A  receive  1 

18.  Bought  the  following  bill  of  goods  of  Mr.  Merchant:, 
what  did  the  whole  amount  to,  N.  Y.  currency  1 

12^  yards  of  cambric  at  Is.  Ad.  per  yard. 

8       "           ribbon  "  2s.  &d. 

21       "            calico  "  Is.  3d. 

6       "            alpaca  "  5s.  Gd. 

4    gallons     molasses  *'  3s.  5d.  per  gallon. 

2^  pounds     tea  "  6s.  6c?.  per  pound. 

30       "           sugar  "  9d.     " 

19.  If  ^  of  a  yard  of  cloth  cost  $3,20,  what  will  i|  of  a 
yard  cost  ] 

Analysis. — Since  5  eighths  of  a  yard  of  cloth  costs  $3,20. 1  eighth 
of  a  yard  will  cost  \  of  $3,20 ;  and  1  yard,  or  8  eighths,  will  cost 
8  times  as  much,  or  f  of  $3,20  ;  ^f  of  a  yard  will  cost  ^|  as  much 
as  1  yard,  or  \^  of  f  of  $3,20 =$4.80. 


160     J 


ANALYSIS.  206 

OPERATION, 

3  <fc«^^M^'60 


m0X^X^X~l  =  U,8Q.     Or,       ,     0 


$ 


I  $4,80. 

20.  If  3|^  pounds  of  tea  cost  3^  dollars,  what  will  9  pounds 
cost  1 

Note. — Reduce  the  mixed  numbers  to  improper  fractions,  and 
then  apply  the  same  mode  of  reasoning  as  in  the  preceding  ex- 
ample. 

21.  \yhat  will  81  cords  of  wood  cost,  if  2f  cords  cost  7-1 
dollars  ? 

22.  If  6  men  can  build  a  boat  in  120  days,  how  long  will 
it  take  24  men  to  build  it  ? 

Analysis. — Since  6  men  can  build  a  boat  in  120  days,  it  will 
take  1  man  6  times  120  days,  or  720  days,  and  24  men  can  build 
it  in  ^  of  the  time  that  1  man  will  require  to  build  it,  or  ^  of  6 
times  120,  which  is  30. 

OPERATION. 


120  X  6-^24  =  30  days.     Or,       . 


30 


Ans.  I  30  days. 

23.  If  7  men  can  dig  a  ditch  in  21  days,  how  many  men 
v/ill  be  required  to  dig  it  in  3  days  ? 

24.  In  what  time  will  12  horses  consume  a  bin  of  oats, 
that  will  last  21  horses  6^  weeks  ? 

25.  A  merchant  bought  a  number  of  bales  of  velvet,  each 
containing  129i|-  yards,  at  the  rate  of  7  dollars  for  5  yards, 
and  sold  them  at  the  rate  of  11  dollars  for  7  yards  ;  and 
gained  200  dollars  by  the  bargain  :  how  many  bales  were 
fiiere  ? 

Analysis. — Since  he  paid  7  dollars  for  5  yards,  for  1  yard  he 
paid  ^  of  S7  or  |  of  1  dollar  ;  and  since  he  received  11  dollars  for 
7  yards,  tor  1  yard  he  received  if-  of  1 1  dollars  or  ^  of  1  dollar. 
He  gained  on  1  yard  the  difference  between  ^  and  ^=-^  of  a  dol- 
lar. Since  his  whole  gain  was  200  dollars,  he  had  as  many  yards 
as  the  gain  on  one  yard  is  contained  times  in  his  whole  gain,  or 
as  ^  is  contained  times  in  200.  And  there  were  as  many  bales 
as  129^.  (the  number  of  yards  in  one  bale),  is  contained  times  in 
the  whole  number  of  yards  ^^  ;  which  givey  9  bale^. 


206 


ANALYSIS. 


OPERATION. 


1291^=1  ^|y",  number  of  yards  in  a  ba.e 


200 


-7-/5 ——^,  whole  number  of  yards  '•  i:00 


Xo.oo_^.35o.o^9  t^ieg^ 


m 


A7rs.     I  9  /;a/cs. 

26.  Suppose  a  number  of  bales  of  cloth,  each  coutaininj^ 
133^  yards,  to  be  bought  at  the  rate  of  12  yards  ibr  1 1  doJ 
lars,  and  sold  at  the  rate  of  8  yards  for  7  dollars,  and  the 
loss  in  trade  to  be  $100  :  how  many  bales  are  there? 

27.  if  a  piece  of  cloth  9  feet  long  and  3  feet  wide,  contain 
3  square  yards  ;  how  long  must  be  a  })iece  of  cloth  that  is  2| 
feet  wide  be,  to  contain  the  same  number  of  yards? 

28.  A  can  mow  an  acre  of  grass  in  4  hours,  B  in  6  hours, 
and  C  in  8  hours.  How  many  days,  worknig  9  hours  a  day, 
would  they  require  to  mow  39  acres  ? 

Analysis. — Suice  A  can  moNv  an  acre  in  4  hours,  B  in  6  hours, 
and  C  in  8  liours,  A  can  mow  ^  of  an  acre,  B  ^  of  an  acre,  and 
C  ^  of  an  acre  in  1  hour.  Together  they  can  mow  i  +  ^+i=-|| 
of  an  acre  in  1  hour.  And  since  they  can  mow  13  twcnty-louriiis 
of  an  acre  in  1  hour,  they  can  mow  1  twenty-fourth  of  an  acre 
in  ^  of  1  hour;  and  1  acre,  or  f^,  in  24  times  ^rr||  of  1  hour: 
and  to  mow  39  acres,  they  will  require  39  times  \^=  \^  hours, 
which  reduced  to  days  of  9  hours  each,  gives  8  days. 


OPERATION. 


\H+\ 


:^  hours. 


u 


_X-X^=..days.  Or 


3 


8 


$  Ans.  I  8  days. 

29.  A  can  do  a  piece  of  work  in  4  days,  and  B  can  do  the 
same  in  6  days  ;  in  what  time  can  they  both  do  the  work  if 
they  labor  together  ? 

30.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  it  ? 

Analysis. — If  6  men  can  do  a  piece  of  work  in  10  day.s,  1  man 
will  require  6  times  as  long,  or  60  days  to  do  the  ^ame  work. 
Five  men  will  require  but  one-fifLli  as  long  as  one  man.  or  60- ^-a 
=  12  days. 


ANALYSlt.  207 

OPERATION. 


10x6-^-5=12  days. 


Ans. 


6 


12  days. 


31.  Three  men  together  can  perform  a  piece  of  work  in  9 
days.  A  alone  can  do  it  in  18  days,  B  in  27  days  ;  in  what 
time  can  0  do  it  alone  ? 

32.  A  and  B  can  build  a  wall  on  one  side  of  a  square 
piece  of  ground  hi  3  days  ;  A  and  C  in  4  days  ;  B  and  C  in 
6  days  :  what  lime  will  they  require,  working  together,  to 
complete  the  wall  endoaing  the  square  1 

33.  Three  men  hire  a  pasture,  for  which  they  pay  66  dol- 
lars. The  first  puts  in  2  liorses  3  weeks  ;  the  second  6  horses 
for  2^  weeks  ;  the  third  9  horses  for  1 J  weelvs  :  how  much 
ought  each  to  pay  % 

Analysis. — The  pasturage  of  2  horses  for  3  weeks,  would  be  the 
same  as  the  pasrurage  of  1  horse  2  times  3  weeks,  or  6  weeks  • 
that  of  six  horses  2-|  weeks,  the  same  as  for  1  hon^e  6  times  2-J 
weeks,  or  15  weeks  :  and  that  of  9  lioryes  1^  weeks,  the  same  as 
1  horse  for  9  times  1^  weeks,  or  12  weeks.  The  tiiree  persons  had 
an  equivalent  for  the  pasiurage  of  I  horse  for  6  +  1 5  + 1 2=  33  weeks ; 
therefore,  the  first  must  pay  ^,  the  second  ^.  and  the  third 
^  of  66  dollars. 

OPERATION. 

3    x2=:6;         then       %%%X^  —  %\2.     1st. 
2^X6  =  15;  "  $66xif:i^S30.     2d. 

11x9  =  12;  "  $66xi|:^$24.     3d. 

34.  Two  persons,  A  and  B.  enter  into  partnership,  and  gain 
$175.  A  puts  in  75  dollars  for  4  months,  and  B  puts"  in  100 
dollars  for  6  months  :  what  is  each  one's  share  of  the  gain  ? 

35.  Three  men  engage  to  build  a  house  for  580  dollars. 
The  first  one  employed  4  hands,  the  second  5  hands,  and  the 
third  7  hands.  The  first  man's  hands  worked  three  times  as 
many  days  as  the  third,  and  the  second  man's  hands  twice  as 
many  dnys  as  the  third  man's  hands  :  how  much  must  each 
receive  % 


208 


ANALYSIS. 


36.  If  8  students  spend  8192  in  6  months,  how  much  will 
12  students  spend  in  20  months  ? 

Analysis. — Since  8  students  spend  $192,  one  student  will  ppend 
\  of  $192.  in  6  months;  in  1  month  1  student  will  spend  ^  of  \ 
of  $192=  $4.  Twelve  students  will  spend,  in  1  month,  12  times 
as  much  as  1  student,  and  in  20  months  they  will  spend  20  times 
as  much  as  in  1  month. 

OPERATION. 


24  2 

un  1  1  ^^  20  ^^^^ 


m 


20 


48 


$960.  Ans. 


37.  If  6  men  can  build  a  wail  80  feet  long,  6  feet  wide, 
and  4  feet  high,  in  15  days,  in  what  time  can  18  men  build 
one  240  feet  long,  8  feet  wide,  and  6  leet  high? 

Analysis. — Since  it  takes  6  men  15  days  to  build  a  wall,  it 
will  take  1  man  6  times  15  days,  or  90  days,  to  build  the  same 
wall.  To  build  a  wall  1  foot  long,  will  require  -^  as  long  as  to 
build  one  80  feet  long ;  to  build  one  1  foot  wide,  \  as  long  as  to 
build  one  4  feet  wide;  and  to  build  one  1  foot  high,  \  as  long  as 
to  build  one  6  feet  high.  18  men  can  build  the  same  wall  in  ^ 
of  the  time  that  one  man  can  build  it:  but  to  build  one  240  feet 
long,  will  lake  them  240  times  as  long  as  to  build  one  1  foot  in 
length;  to  build  one  8  feet  wide,  8  times  as  long  as  to  build  one 
1  foot  wide,  and  to  build  one  6  feet  high,  6  times  as  long  as  to 
build  one  1  foot  high- 


operation. 

*      2 

15x0      1       11       1      ^^0      $     0     ^^ 
^T-^XT-7X-X:rX  — X^  X7X7  =  30. 

1       $0    ^    0    -i^     1     a:    1 


Ans. 


15 

$  2 


30  days. 


38.  If  96lbs.  of  bread  be  sufficient  to  serve  5  men  12  dayp, 
how  many  days  will  Glib,  serve  19  men? 


ANAL 

39.  If  a  man  travel  220  miles' 
hours  a  day,  in  how  many  days  will 
travelling  16  hours  a  day'^ 

40.  If  a  family  of  12  persons  consume  a  certain  quantity 
of  provisions  in  6  days,  how  long  will  the  same  provisions 
last  a  family  of  8  persons  ? 

41.  If  9  men  pay  $135  for  5  weeks'  hoard,  how  much 
must  8  men  pay  lor  4  weeks'  board  i 

42.  If  10  bushels  of  wheat  are  equal  to  40  bushels  of 
corn,  and  28  bushels  of  corn  to  66  pounds  of  butter,  and  39 
pounds  of  butter  to  1  cord  of  wood ;  how  much  wheat  is  12 
cords  of  wood  worth  ? 

Analysis. — Since  10  bushels  of  wheat  are  worth  40  bushels  of 
corn.  1  bushel  of  corn  is  worth  ^  of  10  bushels  of  wheat,  or 
\  of  a  bushel ;  28  bushels  are  worth  28  times  ^  of  a  bushel  of 
wheat,  or  7  bushels :  since  28  bushels  of  corn,  or  7  bushels  of 
wheat  are  worth  56  pounds  of  butter,  1  pound  of  butter  is  worth 
^  of  7=-J  of  a  bushel  of  wheat,  and  39  pounds  are  worth  30 
times  as  much  as  1  pound,  or  39x|=^  bushels  of  wheat;  and 
since  39  pounds  of  butter,  or  -^  busliels  of  wheat  are  worth  1  cord 
of  wood,  12  cords  are  worth  12  tunes  as  much,  or  12X^=c58^ 
bushels. 


OPERATION. 

3  ^    .. 

1        40         1        00         1  1  ^ 

4  2 


n 

39  „ 


2  I  in=:58^bush. 

Note. — Always  commence  analyzing  from  the  term  which  is 
of  the  same  name  or  kind  as  the  required  answer. 

43.  If  35  women  can  do  as  much  work  as  20  hoys,  and 
16  boys  can  do  as  much  as  7  men  :  how  many  women  can 
do  the  work  of  18  men  ? 

44.  If  36  shillings  in  New  York  are  equal  to  27  shillings 
in  Massachusetts,  and  24  shillings  in  Massachusetts  are  equal 
to  30  shillings  in  Pennsylvania,  and  45  shillings  in  Pennsyl- 
vania are  equal  to  28  shillings  in  Georgia  ;  how  many  shil- 
liutjs  in  Georgia  are  equal  to  72  shillings  in  New  York  ? 

14 


210  pKOMisououa  examples 

PROMISCUOUS    EXAMPLES    IN    ANALYSIS. 

1.  How  many  sheep  at  4  dollars  a  head  must  I  give  fei  6 
cows,  worth  1  2  dollars  apiece  '? 

2.  If  7  yards  of  cloth  cost  |49,  what  will  16  yards  cost  ] 

3.  If  36  men  can  build  a  house  in  16  days,  how  long  will 
it  take  12  men  to  build  it '] 

4.  If  3  pounds  of  butter  cost  71  shillings,  what  will  12 
pounds  cost  ? 

5.  If  6^  bushels  of  potatoes  cost  S2|,  how  much  will  1  2-J 
bushels  cost  ? 

6.  How  many  barrels  of  apples,  worth  12  shillings  a  barrel, 
will  pay  for  16  yards  of  cloth,  worth  96'.  6d.  a  yard  ? 

7.  If  31i  gallons  of  molasses  are  w^orth  $9|^,  what  are  5 J 
gallons  worth'? 

8.  What  is  the  value  ol'  24 J  bushels  of  corn,  at  5s.  Id.  a 
bushel,  New  York  currency  ? 

9.  How  much  rye,  at  8s.  3d.  per  bushel,  must  be  given 
for  40  gallons  of  whisky,  worth  2s.  9d.  a  gallon'? 

10.  If  it  take  44  yards  of  carpeting,  that  is  IJ  yards  wide, 
to  cover  a  floor,  how  many  yards  of  |-  yards  wide,  will  it 
take  to  cover  the  same  floor  ? 

11.  If  a  piece  of  wall  paper,  14  yards  long  and  IJ  feet 
wide,  will  cover  a  certain  piece  of  wall,  how  long  must  an- 
other piece  be,  that  is  2  feet  wide,  to  cover  the  same  wall  ? 

12.  If  5  men  spend  $200  in  160  days,  how  long  will  $300 
last  12  men  at  the  same  rate  ? 

13.  If  1  acre  of  land  cost  1  off  of  f  of  $50,  what  will  3  J 
acres  cost  ? 

14.  Three  carpenters  can  finish  a  house  in  2  months  ;  two 
of  them  can  do  it  in  2i  months  :  how  long  will  it  take  tht 
third  to  do  it  alone  1 

15.  Three  persons  bought  2  barrels  of  flour  ibr  15  dollars 
The  first  one  ate  from  them  2  months,  the  second  3  months 
and  the  third  7  months  :  how  much  should  each  pay  ? 

16.  What  quantity  of  beer  will  serve  4  persons  18|  days 
if  6  persons  drink  7^  gallons  in  4  days  1 


IN    ANALYSIS.  211 

17.  If  9  persons  use  l^  pounds  of  tea  in  a  month,  how 
much  will  10  persons  use  in  a  year  ? 

18.  If  ^  of  I  oi"  a  gallon  of  wine  cost  f  of  a  dollar,  what 
will  5^  gallons  cost  ? 

19.  How  many  yards  of  carpeting,  1|  yards  wide,  will  it 
take  to  cover  a  floor  that  is  4|^  yards  wide  and  6  and  three- 
fifths  yards  long  ? 

20.  Three  persons  bought  a  hogshead  of  sugar  containing 
413  pounds.  The  first  paid  $2^  as  often  as  the  second  paid 
^oj,  and  as  often  as  the  third  paid  $4  :  what  was  each  one's 
share  of  the  sugar  ? 

21.  A,  Avith  the  assistance  of  B,  can  build  a  wall  2  feet 
wide,  3  feet  high,  and  30  feet  long,  in  4  days  ;  but  with  the 
assistance  of  C,  they  can  do  it  in  3^  days  :  in  how  many  days 
can  C  do  it  alone  ? 

22.  If  two  persons  engage  in  a  business,  where  one  advances 
$875,  and  the  other  $625,  and  they  gain  $300,  what  is  each 
one's  share  1 

23.  A  person  purchased  ^  of  a  vessel,  and  divided  it  into  5 
equal  shares,  and  sold  each  of  those  shares  for  $1200  :  what 
was  the  value  of  the  whole  vessel  t 

24.  How  many  yards  of  paper,  J  of  a  yard  wide,  will  be 
Buflicient  to  paper  a  room  10  yards  square  and  3  yards  high  ? 

25.  What  will  be  the  cost  of  Adlba.  of  cofiee,  New  Jersey 
currency,  if  9/66'.  cost  27  shillings  1 

26.  What  will  be  the  cost  oi"  o  barrels  of  sugar,  each  weigh- 
ing 2cwf.  at  lOd.  per  pound,  Illinois  currency? 

27.  If  12  mxcn  reap  80  acres  in  6  days,  in  how  many  days 
will  25  men  reap  200  acres  ? 

28.  If  4  men  are  paid  24  dollars  for  3  days'  labor,  how 
many  men  may  be  employed  16  days  for  $96  ? 

29.  If  $25  will  supply  a  family  with  flour  at  $7,50  a  bar- 
rel for  2|  months,  how  long  would  $45  last  the  same  family 
when  flour  is  worth  $0,75  per  barrel  l 

30.  A  wall  to  be  built  to  the  height  of  27  feet,  was  raised 
to  the  height  of  9  feet  by  1 2  men  in  6  days  :  how  many  men 
must  be  employed  to  finish  the  wall  in  4  days  at  the  same 
Kite  of  workin<r  ^ 


212  PROMISCUOUS    EXAMPLES. 

31.  A,  B  and  C,  sent  a  drove  of  hogs  to  market,  of  which 
A  owned  105,  B  75,  and  C  120.  On  the  way  60  died  : 
how  nriany  must  each  lose  ? 

32.  Three  men,  A,  B  and  C,  agree  to  do  a  piece  of  work, 
for  which  they  are  to  receive  $315.  A  works  8  days,  10 J 
hours  a  day  ;  B  9-J  days,  8  hours  a  day  ;  and  C,  4  days,  12 
hours  a  day  :  what  is  each  one's  share  ? 

33.  If  10  barrels  of  apples  will  pay  for  5  cords  of  wood, 
and  12  cords  of  wood  for  4  tons  of  hay,  how  many  barrels  of 
apples  will  pay  for  9  tons  of  hay  ? 

34.  Out  of  a  cistern  that  is  J  full  is  drawn  140  gallons, 
when  it  is  Ibund  to  be  y  lull  :  how  much  does  it  hold  ? 

35.  If  .7  of  a  gallon  of  wine  cost  S2,25,  what  will  .25  of  a 
gallon  cost? 

36.  If  it  take  5.1  yards  of  cloth,  1.25  yards  wide,  to  make  a 
gentleman's  cloak,  how  much  surge,  J  yards  wide,  will  be 
required  to  hne  it? 

37  A  and  B  have  the  same  income.  A  saves  ^  of  his 
annually  ;  but  B,  by  spending  !;?200  a  year  more  than  A,  at 
the  end  of  5  years  finds  himself  $160  in  debt :  w^hat  is  their 
income  ? 

38.  A  father  gave  his  younger  son  $420,  which  was  J  of 
what  he  gave  to  his  elder  son  ;  and  3  times  the  elder  sun's 
portion  was  J  the  value  of  the  father's  estate  :  what  was  the 
value  of  the  estate  ? 

39.  Divide  $176,40  among  3  persons,  so  that  the  first  shall 
have  twice  as  much  as  the  second,  and  the  third  three  times 
as  much  as  the  first  :  what  is  each  one's  share  1 

40.  A  gentleman  having  a  purse  of  money,  gave  ^  of  it  foi 
a  span  of  horses  ;  j  of  ^  of  the  remainder  for  a  carriage  ; 
when  he  found  that  he  had  but  $100  left :  how  much  was  in 
his  purse  before  any  was  taken  out  ? 

41.  A  merchant  tailor  bought  a  number  of  pieces  of  cloth, 
each  containing  25^g  yards,  at  the  rate  of  3  yards  for  4  dol- 
lars, and  sold  them  at  the  rate  of  5  yards  for  13  dollars,  and 
gained  by  the  operation  96  dollars  :  how  many  pieces  did  he 
buy  ^ 


BATIO   AND   PKOPORTION.  213 


RATIO    AND    PROPORTION. 

221.  Two  numbers  having  the  same  unit,  may  be  com- 
pared in  two  ways : 

Ist.  By  considering  how  muck  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference  ;  and, 

2d.  By  considering  how  many  times  one  is  contained  in  the 
other,  which  is  shown  by  their  quotient. 

In  comparing  two  numbers,  one  with  the  other,  by  means 
of  their  difference,  the  less  is  always  taken  from  the  greater. 

In  comparing  two  numbers,  one  with  the  other,  by  means 
of  their  quotient,  one  of  them  must  be  regarded  as  a  standard 
which  measures  the  other,  and  the  quotient  which  arises  by 
dividing  by  the  standard,  is  called  the  ratio. 

222.  Every  ratio  is  derived  from  two  terms:  the  first  is 
called  the  antecedent,  and  the  second  the  consequent ;  and  the 
two,  taken  together,  are  called  a  couplet.  The  antecedent  will 
be  regarded  as  the  standard. 

If  the  numbers  3  and  12  be  compared  by  their  difference, 
the  result  of  the  comparison  will  be  9  ;  for,  12  exceeds  3  by  9. 
If  they  are  compared  by  means  of  their  quotient,  the  result 
will  be  4;  lor,  3  is  contained  in  12,  4  times:  that  is, 
3  riLeasuring  12,  gives  4. 

223.  The  ratio  of  one  number  to  another  is  expressed  in 
two  ways  : 

Ist.  By  a  colon  ;  thus,  3:12;  and  is  read,  3  is  to  12  ;  or, 
3  measuring  12. 

i2 
2c?.  In  a  fractional  form,  as  ~  ;  or,  3  measuring  12. 

o 

221.  In  how  many  ways  may  two  numbers,  having  the  same  unit,  t>e 
compared  with  each  other  1  If  you  compare  by  their  difference,  how  do 
you  find  it  1  If  you  compare  by  the  quotient,  how  do  you  regard  one  of 
the  numbers  ''-     What  is  the  ratio  1 

222.  From  how  many  terms  is  a  ratio  derived  ]  What  is  the  first 
tcnn  called  1     What  is  the  second  called  1     Which  is  the  standard  1 

'VIA    How  limy  tlie  ratio  of  two  nurubcis  be  expressed  ?     Ilinv  re^j*!  ' 


214  RATIO    AND    PROPORTION. 

224.  If  two  couplets  have  the  same  ratio,  their  terms  are 
said  to  be  proportional  :  the  couplets 

3     :     12     and     r    :     4 

have  the  same  ratio  4 ;  hence,  the  terms   are  proportional, 
and  are  written, 

3     :     12     :     :     1     :     4 

by  simply  placing  a  double  colon  between  the  couplets.     The 
terms  are  read 

3  is  to  12       as       1  is  to  4, 
and  taken  together,  they  are  called  a  proportion  :  hence, 

A  proportion  is  a  comparison    of  the   terms  of  two  equal 
ratios.^ 


224.  If  two  couplets  have  the  same  ratio,  what  is  said  of  the  tenns  ( 
Hov/  are  they  written  1     How  read  1     What  is  a  proportion  ? 

*  Some  authors,  of  high  authority,  make  the  consequent  the  stand- 
ard and  divide  the  antecedent  by  it  to  determine  the  ratio  of  the  couplet. 

The  ratio  3  :  12  is  the  same  as  that  of  1:4  by  both  methods  ; 
for,  if  the  antecedent  be  made  the  standard,  the  ratio  is  4  ;  if  the  conse- 
quent be  made  the  standard,  the  ratio  is  one-fourth.  The  question  is, 
which  method  should  be  adopted  1 

The  unit  1  is  the  number  from  which  all  other  numbers  are  derived, 
and  by  which  they  are  measured. 

The  question  is,  how  do  we  most  readily  apprehend  and  express  the 
relation  between  1  and  4  \  Ask  a  child,  and  he  will  answer,  "  the  dif- 
ference is  3  "  But  when  you  ask  him,  "  how  many  I's  are  there  in 
4  1"  he  will  answer,  "  4."  using  1  as  the  standard. 

Thus,  we  begin  to  teach  by  using  the  standard  1  :  that  is,  by  dividing 
4  by  1. 

Now,  the  relation  between  3  and  12  is  the  same  as  that  between  1 
and  4  ;  if  then,  we  divide  4  by  1,  we  must  also  divide  12  by  3.  Do  we, 
indeed,  clearly  apprehend  the  ratio  of  3  to  12,  until  we  have  referred  to 
1  as  a  standard  !  Is  the  mind  satisfied  until  it  has  clearly  perceived  that 
the  ratio  of  3  to  12  is  the  same  as  that  of  1  to  4  ? 

In  the  Rule  of  Three  we  always  look  for  the  result  in  the  4th  term 
Now,  if  we  wish  to  tind  the  ratio  of  3  to  12,  by  referring  to  1  as  a  stand 
ard,  we  have 

3     :      12     :     :     1      :     ratio, 

which  brings  the  result  in  the  right  place. 

But  if  we  define  ratio  to  be  the  antecedent  divided  by  the  consequent, 
we  should  have 

3     :      12     :     :     ratio     :      1, 
wliicb  w(juKl  brinji  tbe  ratio,  or  icqnvcd  number,  in  the  .'3d  jilacc. 


RATIO    AND    PROPORTION. 


215 


What  are  the  ratios  of  the  proportions, 


3 

9 

:        12 

36? 

2 

:       10 

:       12 

60? 

4 

2 

:         8 

4? 

9 

1 

:       90 

10? 

225.  The  1st  and  4th  terms  of  a  proportion  are  caJl^rl  the 
extremes :  the  2d  and  3d  terms,  the  means.  Thus,  in  tb*»  pr»i 
portion, 

3     :     12     :     :     6     :     24 


Since  (Art.  224), 


3  and  24  are  the  extremes,  and  12  and  6  the  means: 
12_24 
3"~"6"' 

we  shall  have,  by  reducing  to  a  common  denominator, 
12x6     24x3 
~3x6~"6^' 

But  since  the  fractions  are  equal,  an(J  have  the  same  deno- 
minators, their  numerators  must  be  equal,  viz  ; 

12x6  =  24x3;  that  is, 

In  any  proportion^  the  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

Thus,  in  the  proportions, 

1   :     6  :   :     2  :   12;  we  have   1x12=   2x6; 
4  :   12  :  :     8  :  24 ;     "       "     4x24=12x8. 

226.  Since,  in  any  proportion,  the  product  of  the  extremes 
\&  equal  to  the  product  of  the  means,  it  follows  that, 

In  all  cases,  the  numerical  value  of  a  (juantity  is  the  number  of  time? 
which  that  quantity  contains  an  assumed  standard,  called  its  unit  oj 
mea»  ure. 

If  we  would  find  that  numerical  value,  in  its  right  place,  we  musl 
3ay» 

standard     :     quantity     :     :      1      :     numerical  value  : 

but  if  we  take  the  other  method,  we  have 

quantity  standard     :  numerical  value     :      1. 

which  brings  the  iiumerioiil  value  in  the  wrotiir  place. 


2^16  RATIO    AND    PKOPOiJTION. 

Isti^  If  the  product  jof  the  means  be  divided  by  one  of  ike 
extremes,  the  quotient  will  he  the  other  extreme. 

Thus,  in  the  pioportion 

3   :   12  :  :  6  :  24,  we  have    3x24  =  12x6; 

then,  if  72,  the  product  of  the  means,  be  divided  by  one  o* 
the  extremes,  3,  the  quotient  will  be  the  other  extreme,  24: 
or,  if  the  product  be  divided  by  24,  the  quotient  will  be  3. 

2d.  If  the  product  of  the  extremes  be  divided  by  either  of 
the  means,  the  quotient  will  be  the  other  mean. 

Thus,  if  3  X  24  irr  12  X  6:=  72  be  divided  by  12,  the  quotient 
will  be  6  ;  or  if  it  be  divided  by  6,  the  quotient  will  be  12. 

EXAMPLES. 

1.  The  first  three  terms  of  a  proportion  are  3,  9  and  12  : 
"what  is  the  fourth  term  ] 

2;  The  first  three  terms  of  a  proportion  are  4,  16  and  15  : 
what  is  the  4th  term  1 

3.  The  first,  second,  and  fourth  terms  of  a  proportion  are 
6,  12   and  24  :   what  is  the  third  term  ? 

4.  The  second,  third,  and  fourth  terms  of  a  proportion  are 
9,  6   and  24  :  what  is  the  first  term.^ 

5.  The  first,  second  and  fourth  terms  are  9,  18  and  48  ; 
what  is  the  third  term? 

227.   Simple  and  Compound  Ratio. 

The  ratio  of  two  single  numbers  is  called  a  Simple  Ratio^ 
and  the  proportion  which  arises  from  the  equality  of  two  such 
ratios,  a  Simjjle  Proportion. 

225.  Which  are  th.;  extremes  of  a  proportion  1.  Which  the  means  1 
What  is  the  product  of  the  extremes  equal  to  1 

226.  If  the  product  of  the  means  be  divided  by  one  of  the  extremes, 
what  will  the  quotient  be  1  If  the  product  of  the  means  be  divided  by 
either  extreme,  what  will  the  quotient  be  ? 

227.  What  is  a  simple  ratio  ?  What  is  the  proportion  called  whicli 
comes  from  the  equality  of  two  simple  ratios  1  What  is  a  conipoiuul 
ratio  '     What  is  u  compound  pruporliuu  ' 


eiMl'LE 

QUESTIO 

1.  If  12  apples  be  equally  dividea  aiiiUliy  4  boys,  how 
many  will  each  have  ? 

Analysis. — Since  12  apples  are  to  be  divided  equally  between 
4  boys,  one  boy  will  have  as  many  apples  as  4  is  contained  timeh 
In  12,  which  is  3  ;  therefore,  if  12  apples  be  equally  divided  be- 
tween 4  boys,  each  will  have  3  apples. 

2.  If  24  peaches  be  equally  divided  among  6  boys,  how 
many  will   each  have  ?     How  many  times  is  6   contained  in 

3.  A  man  has  32  mil&i  to  walk,  and  can  travel  4  miles  an 
:  our,  how  many  hours  will  it  take  him  ? 

4.  How  many  yards  of  cloth,  at  3  dollars  a  yard,  can  you 
buy  for  24  dollars  ? 

Analysis. — Since  the  cloth  is  3  dollars  a  yard,  you  can  buy  as 
many  yards  as  3  is  contained  times  in  24,  which  is  8  :  therefore, 
you  can  buy  8  yards. 

5.  How  many  oranges  at  6  cents  apiece  can  you  buy  for 
i  2  cents  ? 

6.  How  many  pine-apples  at  12  cents  apiece  can  you  buy 
.or  132  cents'? 

7.  A  farmer  pays  28  dollars  for  7  sheep  :  how  much  is 
that  apiece? 

Analysis. — Since  7  sheep  cost  28  dollars,  one  sheep  will  cost  ass 
many  dollars  as  7  is  contained  times  in  28,  which  is  4  ;  therefore, 
each  sheep  will  cost  4  dollars. 

8  If  12  yards  of  muslin  cost  96  cents,  how  much  does 
1  yard  cost] 

9.  How  many  lead  pencils  could  you  buy  for  42  cents,  if 
they  cost  6  cents  apiece  ? 

10.  How  many  oranges  could  you  buy  for  72  cents,  if  they 
cost  6  cents  apiece  ? 

1 1 .  A  trader  wishes  to  pack  64  hats  in  boxes,  and  can  put 
but  8  hats  in  a  box  :  how  many  boxes  does  he  want  1 

12.  If  a  man  can  build  7  rods  of  fence  in  a  day,  how  long 
will  it  take  him  to  build  77  rods  ? 

13.  If  a  man  pays  t>G  dollars  for  seven  yards  of  cloth,  how 
much  i«  that  a  yard  ? 


68 


DIVISION. 


14.  Twelve  men  receive  108  dollars  for  doing  a  piece  of 
work  :  how  much  does  each  one  receive '? 

15.  A  merchant  has  144  dollars  with  which  he  is  going  to 
buy  cloth  at  12  dollars  a  yard  ;  how  many  yards  can  he  pur- 
chase ? 

16.  James  is  to  learn  forty-two  verses  of  Scripture  in  a 
week  :  how  much  must  he  learn  each  day  ? 

17.  How  many  times  is  4  contained  in  50,  and  how  maii) 
over  'i 

PRINCIPLKS    AND    EXAMPLES- 

60.   1.  Let  it  be  required  to  divide  86  by  2. 

Set  down  the  number  to  be  divided  and  write 
the  other  number  on  the  left,  drawing  a  curved 
line  between  them.  Now  there  are  8  tens  and 
6  units  to  be  divided  by  2.  We  say,  2  in  8.  4 
timeS;  which  being  tens,  we  write  it  in  the  tens 
place.  We  then  say,  2  in  6,  3  times,  which 
being  units,  are  written  in  the  units'  place. 
The  result,  which  is  called  a  quotient,  is  fcheie- 
lore,  4  tens  and  3  units,  or  43. 


OPERATION. 

n3 


> 

86 

43  quotie't. 


OPVl'HATIOJI. 

;^)729 
84^ 


2.  Let  it  be  required  to  divide  729  by  3. 
Analysis. — We  say,  3  in  7,  2  times  and  1  over. 

Set  down  the  2,  which  are  hundreds,  under  the  7. 

But  of  the  7  hundreds  there  is  i  hundred,  or  10  tens, 

not  yet  divided.     We  pat  the  10  tens  with  the  2 

tens,  making  12  tens,  and  then  say,  3  in  12,  4  times,  and  write  the 

4  of  the  quotient  in  the  tens'    place ;  then   say,  3  in  9,  3    times. 

The  quotient,  therefore,  is  243. 

3.  Let  it  be  required  to  divide  466  by  8. 
Analysis. — We  first  divide  the  46  tens 

by  8,  giving  a  quotient  of  5  tens,  and  6  tens 
over.  These  6  tens  are  equal  to  60  units, 
to  which  we  add  the  6  in  the  units'  place. 
We  then  say,  8  in  66,  8  times  and  2  over ; 
hence,  the  quotient  is  58,  and  2  over,  wliich 
we  call  a  remainder.  This  remainder  is 
written  after  the  last  quotient  figure,  and 
the  8  placed  under  it ;  the  quotient  is  read, 
68  and  2  divided  by  8. 


OPERATION. 

8)466 

08-2  remain. 


58|  quotient. 


CO    Ex.  1.- 
tens  or  uaitsj  1 


■When  you  divide  8  tens  by  2,  is  the  unit  of  the  quotient 
When  G  unite  are  divided  by  %,  what  is  the  unit? 


SIMPIJi   NUMBERS.  59 

Analysis. — In  the  first  example,  86  is  divided  into  2  equal  parts, 
and  the  quotient  43  is  one  of  the  parts.  If  one  of  the  equal  parts 
be  multiplied  by  the  number  of  parts  2,  the  product  wilMae  86,  the 
number  divided. 

In  the  third  example.  466  is  divided  into  8  equal  parts,  and  two 
units  remain  that  are  not  divided.  If  one  of  the  equal  parts.  5vS, 
be  multiplied  by  the  number  of  parts,  8,  and  the  remainder  2  be 
added  to  the  product,  the  result  will  be  equal  to  466,  the  numbej 
divided. 

6 1 .  Division  is  the  operation  of  finding  from  two  numbers 
a  third,  which  multiplied,  by  the  first,  will  produce  the  second. 

The  first  number,  or  number  by  which  we  divide,  is  called 
the  divisor. 

The  second  number,  or  number  to  be  divided,  is  called  the 
dividend. 

The  third  number,  or  result,  is  called  the  gtiotie^it. 

The  quotient  shows  how  many  times  the  dividend  contains 
the  divisor. 

II"  anything  is  left  after  division,  it  is  called  a  remainder. 

62.  There  are  three  parts  in  every  division,  and  sometimes 
four:  1st,  the  dividend;  2d,  the  divisor;  3d,  the  quotient; 
and  4th,  the  remainder. 

There  are  three  signs  used  to  denote  division  ;  they  are  the 
following : 

18-^4  expresses  that  18  is  to  be  divided  by  4. 
Jj^         expresses  that  18  is  to  be  divided  by  4. 
4)18  expresses  that   18  is  to  be  divided  by  4. 
When  the  last  sign  is  used,  if  the  divisor  does  not  excejd 
12,  we  draw  a  line  beneath,  and  set  the  quotient  under  it.  Il 
the  divisor  exceeds  12,  we  draw  a  curved  line  on  the  right  of 
the  dividend,  and  set  the  quotient  at  the  right. 

2. — When  the  seven  hundreds  are  divided  by  3.  what  is  the  unit  of 
the  quotient  1  To  hovy  many  tens  is  the  undivided  hundred  equal  1 
When  the  12  tens  are  divided  by  3,  vs-hat  is  the  unit  of  the  quotient  1 
When  the  9  units  are  divided  by  3,  what  is  the  quotient '! 

3. — How  is  the  division  of  the  remainder  expressed  1  Read  the 
quotient.  If  there  be  a  remainder  after  division,  how  must  it  be  written ! 

61.  What  is  division]  What  is  the  number  to  be  divided  called? 
What  is  the  number  called  by  which  we  divide  I  What  is  the  answci 
called?     What  is  the  number  called  which  is  leftl 

62.  How  many  parts  are  there  in  division  1  Name  them.  IIow 
many  siyns  ure  there  in  divijiimi ''     Makf  and  name  tlieiii 


00  SHORT   DIVISION. 


SHORT  DIVISION. 

« 

63.  Short  Division  is  the  operation  of  dividing  when  the 
work  is  performed  mentally,  and  the  results  only  written 
down.  It  is  limited  to  the  cases  in  which  the  divisors  do  not 
exceed  12. 

Let  it  be  required  to  divide  30456  by  8. 

A.NAi-Ysis. — We  first  fiay,  8  in  3  we  cannot.  Then,  operation. 

8  in  30;  3  times  and  6  over  ;  then,  8  in  64,  8  times  ;  8)30456 

then  8  in  5,  0  times  j  then.  8  in  56,  7  times  :  hence,  — ^S07~ 

Rule  I. —  Writs  the  divisor  on  the  hft  of  the  dividend. 
Beginning  at  iht  left,  divide  each  Jig  are  of  the  dividend  hy 
the  divisor,  and  set  each  quotient  figure  under  its  dividend. 

II.  If  there  is  a.  remainder,  after  ony  diviaion,  annex  to  it 
^Jie  next  figure  of  the  dividend,  and  divide  as  before. 

III.  If  any  dividend  is  less  than  the  divisor,  write  0  for  th, 
quotient  figure  and  annex  the  next  figure  of  the  dividend,  fo 
a  new  dividend. 

iV.  If  there  is  a  remainder,  after  dividing  the  last  figurb^ 
eet  the  divisor  under  it,  and  annex  the  result  to  the  quotient 

Proof. — Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder,  when  there  is  one  ;  if  the  wck 
is  right  the  result  will  be  equal  to  the  dividend 

EXAMPLES. 

•         (1)  (2.)  (3.)  ,4)      . 

3)9369         4)73684  ;)673420         6)325467 

Ans.         3123  "18421  T3~4684  T37577| 

3  4  5  6 


Proof       9369  73684  673420  825^67 

5.  Divide  86434  by  2. 


6.  Divide  416710  by  4. 

7.  Divide  64140  by  5. 

8.  Divide  278943  by  6. 

9.  Divide  95040522  by  6. 

10.  Divide  75890496  by  8. 

11.  Divide  6794108  by  3. 
2.  Divide  21090^131  by  9. 


13.  Divide  2345678964  uy  6 

14.  Divide  570196382  by  12. 

15.  Divide  67897634  by  9. 

16.  Divide  75436298  by  12. 

17.  Divide  674189904  by  9. 

18.  Divide  1404967214  by  U 

19.  Divide  27478041  by  10. 
20  Divid.'  1671  ^'1329  by  12 


FJtACTlONS.  61 

21.  A  man  sold  his  farm  for  6756  dollars,  and  divided  the 
amount  equally  between  his  wife  and  0  children  :  how  much 
did  each  receive  i 

22.  There  are  576  persons  in  a  train  of  12  cars  :  how 
many  are  there  in  each  car  1 

23.  If  a  township  of  land  containing  2304  acres  be  equally 
divided  between  8  persons,  how  many  acres  will  each  have  1 

24.  If  it  takes  5  bushels  of  wheat  to  make  a  barrel  of  flour, 
how  many  barrels  can  be  made  from  65890  bushels  ? 

25.  Twelve  things  make  a  dozen  :  how  many  dozens  are 
there  in  2167284  ? 

26.  Eleven  persons  are  all  of  the  same  age,  and  the  sum 
of  their  ages  is  968  years :  what  is  the  age  of  each  ? 

27.  How  many  barrels  of  flour  at  7  dollars  a  barrel  can  be 
bought  fbr  609463  dollars  1 

28.  An  estate  worth  2943  dollars,  is  to  be  divided  equally 
between  a  lather,  mother,  3  daughters  and  4  sons  :  what  is 
the  portion  of  each  ? 

29.  A  county  contains  207360  acres  of  land  lying  in  9  town- 
ships of  equal  extent :  how  many  acres  in  a  township  ? 

30.  If  11  cities  contain  the  same  number  of  inhabitants, 
and  the  whole  number  is  equal  to  3800247  :  how  many  will 
there  be  in  each  ? 

FRACTIONS. 

64.  1.  If  any  number  or  thing  be  divided  into  two  equal 
parts,  one  of  the  parts  is  called  one-half,  which  is  written 
thus ;  ^. 

2.  If  any  number  is  divided  into  three  equal  parts,  one  of 
the  parts  is  called  one-third,  which  is  written  thus  ;  J ;  two 
of  the  parts  are  called  two-thirds,  and  written  thus  ;  f. 

3.  If  any  number  is  divided  into  four  equal  parts,  one  of 
the  parts  is  called  one-fourth,  which  is  written  thus  ;  \  ;  two 
of  the  parts  are  called  two-fourths,  and  are  written  thus  ;  ^  ; 
three  of  them  are  called  three-fourths,  and  written  |- ;  and 
similar  ntimes  are  given  to  the  equal  parts  into  wdiich  any 
number  may  be  divided. 

03.  What  is  short  division  1  How  is  it  generally  pcrfomicil'  (Jivt 
thf  ruif.     H(»w  tlo  viru  pr(jv<;  i-horl  (]ivi.>-i()i»  1 


62  FKACTIONS. 

4.  If  a  number  "s  divided  into  five  equal  parts,  what  is  one 
of  the  parts  called  ?  Two  of  them  ?  Three  of  them  ]  Four 
ol"  them  ? 

5.  If  a  number  is  divided  into  7  equal  parts,  what  is  one 
of  the  parts  called  ?  What  is  one  of  the  parts  called  when 
it  is  divided  mto  8  equal  parts  ?  When  it  is  divided  into  9 
equal  parts  ?  When  it  is  divided  into  10  ?  When  it  is  divided 
into  11  1     When  it  is  divided  into   12  ? 

6.  What  is  one-half  of  2  ?  of  4  ?  of  6  ?  of  8  ?  of  10  ?  of  12  ? 
of  14?  of  16?  of  18? 

7.  What  is  one-third  of  3  ?     What  is  two-thirds  of  3  ? 

ANA.LYSIS. — Two-thirds  of  three  are  two  times  one-third  of 
three.  One-third  of  three  is  1  ;  therefore,  two-thirds  of  three  are 
two  times  1,  or  2. 

Let  every  question  be  analyzed  in  the  same  manner. 

Wliat  is  one-third  of  6  ?  2  thirds  of  6  ?  One-third  of  9  ? 
2  thuds  of  9  ?     One-third  of  12  ?  two- thirds  of  12  ? 

8.  What  is  one-fourth  of  4  ?  2  fourths  of  4  ?  3  fourths  of  4  ? 
What  is  oiie-fourth  of  8  ?  2  fourths  of  8  ?  3  fourths  of  8  ?  What 
is  onc-fburth  of  12  ?  2  fourths  of  12  ?  3  fourths  of  12  ?  One- 
fourth  of  16  ?  2  fourths  of  16  ?  3  fourths'? 

9.  What  is  one-seventh  of  7  ?  What  is  2  sevenths  of  7  ?  5 
sevenths  ?  6  sevenths?  What  is  one-seventh  of  14  ?  3  sev- 
enths ?  5  sevenths  ?  6  sevenths  ?  What  is  one-seventh  of  21 1 
of  28  ?  of  35  ? 

10.  W^hat  is  one-eighth  of  8  ?  of  16  ?  of  24  ?  of  32  ?  of 
40  ?  of  56  ? 

11.  What  is  one-ninth  of  9  ?  2  ninths?  7  ninths?  6  ninths? 
5  ninths?  4  ninths?  What  is  one-ninth  of  18?  of  27?  of 
64?  of  72?  of  90?  of  108? 

12.  How  many  halves  of  1  are  there  in  2? 

Analysis. — There  are  twice  as  many  halves  in  2  as  there  are 
in  1 ,  There  are  two  halves  in  1  ;  therefore,  there  are  2  times  2 
halves  in  2,  or  4  halves. 

13.  How  many  halves  of  1  are  there  in  3  ?  In  4  ?  In  5? 
In  6?  In  8?  In  10?  In  12? 

14.  How  many  thirds  are  there  in  1  ?  How  many  thirds 
of  1  in  2  ?  In  3  ?  In  4  ?  In  5  ?  In  6  ?  In  9  ?  In  12  ? 

1 5.  How  many  fourths  are  there  in  1  ?  How  many  f.»urths 
i;f  1  in  2?  In  4?   In  6?  In  10  ?  In  12? 


FKAOTIONS.  6S 

16.  How  many  fifths  are  there  in  1  ?     How  many  fifths  oi 

1  are  there  lu  2  V  In  3  ?  In  6  ?  In  7  ?  In  11?   In  12  ? 

17.  How  many  sixths  are  there  in  2  and  one-sixth  i  In  3 
and  4  sixths  ?   In  0  and  2  sixths  1  In  8  and  6  sixths  1 

18.  How  many  sevenths  of  1  are  there  in  2  ?  In  4  and  3 
sevenths  how  many  i     How  many  in  5  and  5  sevenths  ?     In 

and  6  sevenths  ? 

1  9.  How  many  eighths  of  1  are  there  in  2  ?  How  many 
in  2  and  3  eighths  'I  In  2  and  5  eighths  ?  In  2  and  7  eighths  ? 
In  3  ?  In  3  and  4  eighths ?  In  9  ?  In  9  and  5  eighths'?  In 
10^  In  10  and  7  eighths? 

20.  How  many  twelfths  of  1  are  there  in  2"?  In  2  and  4 
twelfths  how  many  ]  How  many  in  4  and  9  twelfths?  How 
many  in  5  and  10  twelfths  ?  In  6  and  9  twelfths  1  In  10  and 
11  twelfths? 

21.  What  is  the  product  of  12  multiplied  by  3  and  one- 
half,  (which  is  written  3^)  ? 

Analysis. — Twelve  is  to  be  taken  3  and  one-half  times  (Art. 
45).  Twelve  taken  i  times  is  6;  and  12  taken  three  times  is  36  j 
therefore,  12  taken  3i  times  is  42. 

22.  What  is  the  product  of  10  multiphed  by  6 J  1 

23.  What  is  the  product  of  12  multiplied  by  3^  ? 

24.  What  is  the  product  of  8  multiplied  by  4^  ? 

25.  What  will  9  barrels  ol'  sugar  cost  at  2|  dollars  a 
bairei  ? 

Analysis. — Nine  barrels  of  sugar  will  cost  nine  times  as 
<nuch  as  1  barrel.  If  one  barrel  of  sugar  costs  2§  dollars,  9 
barrels  will  cost  9  times  2|  dollars,  whiclj   are  24   dollars.     For, 

2  thirds  taken  9  times  gives  18  thirds,  which  are  equal  to  6 ;  then 
9  times  2  are  18,  and  6  added  gives  24  dollars. 

26.  What  will  6  yards  of  cloth  cost  at  5|-  dollars  a  yard  1 

27.  What  will  12  sheep  cost  at  4^  dollars  apiece  1 

28.  W^hat  will  10  yards  of  calico  cost  at  9|  cents  a  yard? 

29.  What  will  8  yards  of  broadcloth  cost  at  7|  dollari 
A  yard  ? 

30.  What  will  9  tons  of  hay  cost  at  9f  dollars  a  ton'? 

31.  How  many  times  is  2-i  contained  in  10  ? 

Analysis. — Two  and  one-half  is  equal  to  5  halves  ;  and  10  ia 
ftqual  to  20  halves ;  then,  5  halves  is  contained  in  20  halves  4 
times  :  hence. 


64  LONG    DIVISION. 

In  all  similar  questions  change  the  divisor  and  dividend 
to  the  same  fractional  unit. 

32.  How  many  yards  of  cloth,  at  3^  dollars  a  yard,  can 
you  buy  for  14  dollars?  how  many  for  21  dollars'? 

33.  If"  oranges  are  3^  cents  apiece,  how  many  can  you  buy 
for  20  cents  % 

34.  If  1  yard  of  ribbon  costs   2 J  cents,  how  many  yards 
can  you  buy  for  12  cents  1 

35.  If  1   yard    of  broadcloth   costs   3|   dollars,  how  many 
yards  can  be  bought  for  33  dollars'? 

36.  If  1  pound  of  sugar  costs  4^  cents,  how  many  pounds 
can  be  .bought  for  36  cents  % 

37.  How  many  times  is  6\  contained  in  44  ? 

38.  How  many  times  is  2|^  contained  in  24  ? 

39.  How  many  lemons,  at   2 J  cents  apiece,  can  you  buy 
for  32  cents  ? 

40.  How  many  yards  of  ribbon,  at  1|^   cents   a   yard,  can 
you  buy  for  12  cents  ? 

LONG  DIVISION. 

65.  Long  Divtston  is  the  operation  of  finding  the  quotient 
of  one  number  divided  by  another,  and  embraces  the  case  of 

Short  Division,  treated  in  Art.  63. 

1.  Let  it  be  required  to  divide  7059  by  13. 

Analysis. — The    divisor,  13,  is  not 

contained    \n    7     thousands ;    therefore,  operation. 

there  are  no  thousands  in  the  quotient.  ^  ^a        .     « 

We  tlie*^    consider  the  0  to  be   annex-  ©  1  2  .-§    'a  **  .^ 

ed   to   the  7,  makmg   70  hundreds,  and  H  S  E^  5    ffi  H  5 
call  this  a  partial  dividend.                               ^  (i\i  (\  ^  a  (  >={  a  'i 

The  divisor,    13,  is  contained    in    70       l^j/uoy^040 

hundreds,    5  hundreds  times   and  some-  ^  " 

thing    over.     To  find    how  much  over,  5  5 

/nultiply  13  by  5  hundreds  and  subtract  5  2 

<he  product  65  from  70,  and  there  will  — o-g 

remain    5    hundreds,    to    which    bring  q 

io>vn    the    5  tens,   and  consider  the  55  *^ 
>ens  a  new  partial  dividend. 

65.  What  is  long  division  !     Docs  it  embrace  the  rase  o*"  short  divi- 

ion  !     What  is  a  partial  dividend  ^ 


SIMPLE   NUMBERS.  66 

Then,  13  is  contained  in  55  tens,  4  tens  times  and  something 
over.  Multiply  13  by  4  tens  and  subtract  the  product,  52,  from 
65,  and  to  tlie  remainder  3  tens  bring  down  the  9  units,  and  con- 
sider the  39  units  a  new  partial  dividend. 

Then,  13  is  contained  in  39,  3  times.  Multiply  13  by  3,  and 
subtract  the  product  39  from  39,  and  we  find  that  nothing  remains. 

66.  Proof. — Each  product  that  has  arisen  from  multiply- 
ng  the  divisor  by  a  flj^ure  of  the  quotient,  is  a  partial  product, 

and  the  sum  of  these  products  is  the  product  of  the  divisor 
and  quotient  (Art.  51,  Note).  Each  product  iias  been  taken, 
separately,  from  the  dividend,  and  nothing  remains.  But, 
taking  each  product  away  in  succession,  leaves  the  same  re- 
mainder as  would  be  left  if  their  sura  were  taken  away  at 
once.  Hence,  the  number  543,  when  multiplied  by  the 
divisor,  gives  a  product  equal  to  the  dividend  :  therefore,  543 
is  the  quotient  (Art.  61)  :  hence,  to  prove  division, 

Multiply  the  divisor  hy  the  quotient  and  add  in  the  remaiyi- 
der,  if  any.  If  the  work  is  right,  the  result  will  be  the  same 
as  the  dividend. 

67.  Let.)it  be  required  to  divide  2756  by  26. 

We  first  say,  26  in  27  once,  and  place  1  in  operation. 

the  quotient.  Multiplying  by  1,  subtracting,  26)2756(106 
and    bringing  down   the  5,  we  have   15  lor  the  26 

first  partial  dividend.     We  then  say,   26  in  15, 
0  times,  and  place  the  0  in  the  quotient.     We 


156 
156 


is  contained  in  156,  6  times. 

If  any  one  of  the  partial  dividends  is  less  than  the  divisor,  write 
0  for  the  quotient  figure,  and  then  bring  down  the  next  figure, 
forming  a  new  partial  dividend. 

Hence,  for  Long  Division,  we  have  the  following 

Rule. — 1.   Write  the  divisor  on  the  left  of  the  dividend. 

II.  Note  the  fewest  figures  of  the  dividend,  au  the  left^ 

that  will  contain  the  divisor,  and  set  the  quotient  figure  at 

the  right. 

66.  What  is  a  partial  product  1  W'hat  is  the  sum  of  all  the  partial 
products  equal  to  (     How  do  you  prove  division  ' 

67.  ^^'hat  do  you  do  if  any  partial  dividend  is  less  than  the  divisor  * 
What  is  the  rule  for  long  division  1 

5 


6(5 


T.Oyr;     T>TVTSTON 


III.  Myltiply  the  divisor  by  the  quntiefit  figitrre,  subtract 
the  'product  from  the  first  'partial  dividend^  and  to  the  re- 
mainder  annex  tlic  next  figure  of  the  dividend^  forming  a 
second,  partial  divideiid. 

IV.  Find  in  tlie  same  manner  the  second  and.  svcceeding 
'figures  of  the  quotient,  till  all  the  figures  of  the  dividend 
are  brought  down. 

NoTK  1. — There  are  five  operations  in  Long  Division.  1st.  To 
write  down  the  numbers :  2d.  Divide,  or  find  how  many  times  : 
3d.  Multiply  :  4th.  Subtract:  5th.  Bring  down,  to  form  the  partial 
dividends. 

2.  The  product  of  a  quotient  figure  by  the  divisor  must  never 
be  larger  than  the  corresponding  partial  dividend  :  if  it  is,  the 
quotient  figure  is  too  large  and  must  be  diminished. 

3.  When  any  one  of  the  remainders  is  greater  than  the  divisor, 
the  quotient  figure  is  too  small  and  must  be  inerea.>^ed. 

4.  The  unit  of  any  quotient  figure  is  the  same  as  that  of  the 
partial  dividejid  from  which  it  is  obtained.  The  pupil  should 
always  name  Lhe  unit  of  every  quotient  figure. 


EXAMPLES. 


1.  Divide  7574  by  54. 

OPERATION. 

64)7574(140 
54 

2\i 
216 

11 
00 
14  Remainder. 

2.  Divide  67289  by  261, 

OPERATION. 

201)67289(257 
522_ 
"1508 
1305 
^039 
1827 
212  Remainder 


PROOF. 

140  Quotient. 
54  Divisor. 

"560" 
700 

7560 

14  Remainder 

7574   Dividend. 

PROOF. 

261   Divisor. 
257  (Quotient 

1827 
1305 
522 

2~r2  Remainder 

67281)  Dividend. 

blMPLK  ^UMISKKS.  67 

3.  Divide  119836687  by  39407. 


OPERATION. 

39407)119836687(304 
118221 

PROOF 

39407   Diviaor. 
3041    duotienl 

161568 
157628 

39407 
39407 

39407 
157628 
118221 
119836687  Dividend. 

4.  Divide  7210473  by  37.    9.  Divide  62015735  by  78. 

5.  Divide  147735  by  45.  10.  Divide  14420946  by  74. 

6.  Divide  937387  by  54,  11  Divide  295470  by  90. 

7.  Divide  145260  by  108.  12.  Divide  1874774  by  162. 

8.  Divide  79165238  by  238.  13.  Divide  435780  by  216. 

14.  Divide  203812983  by  5049. 

15.  Divide  20195411808  by  3012. 

16.  Divide  74855092410  by  949998. 

17.  Divide  47254149  by  4674. 

18.  Divide  119184669  by  38473. 

19.  Divide  280208122081  by  912314. 

20.  Divide  293839455936  bv  8405. 

21.  Divide  4637064283  by  57606. 

22.  Divide  352107193214  by  210472. 

23.  Divide  558001172606176724  by  2708630425. 

24.  Divide  1714347149347  by  57143. 

25.  Divide  6754371495671594  by  678957. 

26.  Divide  71900715708  by  37149. 

27.  Divide  571943007145  by  37149. 

28.  Divide  671493471549375  by  47143 

29.  Divide  571943007645  by  37149. 

30.  Divide  171493715947143  by  57007. 

31.  Divide  121932631112635269  by  987654321. 


Notes. — 1.  How  many  operations  are  there  in  long  division  ?  Nani€ 
them. 

2.  If  a  partial  product  is  greater  than  the  partial  dividend,  what  loet 
it  indicate  '      What  do  you  do  ! 

3.  What  do  you  do  when  any  one  of  the  remainders  is  greater  thMl 
the  divi.sor  ] 

4.  What  is  the  unit  of  any  figure  of  the  quotient  1  When  the  divisoi 
is  contained  in  simple  units,  wliat  will  be  the  unit  of  the  quotient  figure? 
When  it  is  contained  in  tens,  what  will  be  the  unit  of  the  quotient 
figure  1     When  it  is  contained  in  hundreds  ^     In  thousands  ! 


t)i>  i^OiNG    DIVISION. 

68.    PRINCIPLES    RESULTING    FROM    DIVISION. 

Notes. —  1st.  When  the  divisor  is  1,  the  quotient  will  be  equal 
to  tlje  dividend. 

2d.  When  the  divisor  is  equal  to  the  dividend,  the  quotient 
will  be  1. 

3d.  When  the  divisor  is  less  than  the  dividend,  the  quotient 
will  be  greater  than  1.  The  quotient  will  be  as  many  times 
greater  than  1,  as  the  dividend  is  times  greater  than  the  divisor. 

4th.  When  the  divisor  is  greater  than  the  dividend,  the  quotient 
will  be  less  than  1.  The  quotient  will  be  such  a  part  of  I,  as 
the  dividend  is  of  the  divisor. 

PROOF    OF    MULTIPLICATION. 

69.  Division  is  the  reverse  of  multiplication,  and  the)' 
prove  each  other.  The  dividend,  in  division,  corresponds  to 
the  product  in  multiplication,  and  the  divisor  and  quotient  to 
the  multiplicand  and  multiplier,  which  are  factors  of  the  pro- 
duct :  hence, 

If  the  product  of  two  numbers  he  divided  by  the  multipli- 
cand., the  quotient  will  be  the  multiplier  ;  or,  if  it  be  divided 
by  the  multiplier.,  the  quotient  will  be  the  multiplicand. 


EXAMPLES. 

3679  Multiplicand.                  36 

179)1203033 

327  Multiplier. 

11037 

25753 

9933 

7358 

7358 

11037 

25753 

1203033  Product. 

25753 

2.  The     multiplicand     is    61835720,     and    the     product 

8162315040  :  what  is  the  multiplier? 

3.  The    multiplier   is    270000  ;    now    if  the    product  be 
1315170000000,  what  will  be  the  multiplicand? 

4.  The  product  is  68959488,  the  multiplier  96:   what   is 
tlie  multiplicand  ? 

5.  The   multiplier  is    1440,    the   product    10264849920 
what  is  the  multiplicand  ? 

6.  The    product    is    6242102428164,    the    multiplicand 
6795634  :  what  is  the  multiplier  ? 


tX>NTK ACTIONS    IN    MX^LTIPLICATION.  Oil 

CONTRACTIONS   IN   MULTIPLICATION. 

70.  To  multiply  by  25. 

1.  Multiply  275  by  25. 

Analysis. — If  we  annex  two  ciphers  to  the  mnl-      operation. 
tiplicand,  we  multiply  it  by  100   (Art.  55)  :  this       4)27500 
product  is  4  times  too  great;  for  the  multiplier  is  fift75~ 

but  one-fourth  of  100  ;  hence,  to  multiply  by  25, 

Annex  two  ciphers  to  the  multiplicand  and  divide  tJie 
residt  dy  4:. 

EXAMPLES. 

1.  Multiply  127  by  25.       I     3.  Multiply  87504  by  25. 

2.  Multiply  4269  by  25.     |     4.  Multiply  704963  by  25. 

71.  TomuUiply  hy  \2\. 

1.  Multiply  326  by  12i  , 

Analysis. — Since  12^  is  one-eighth  of  100.  operation. 

Annex  tivo  ciphers  to  the  multiplicand  and  di-  8)32600 

vide  the  result  hy  H.  ~^40Vy 

EXAMPLES. 


1.  Multiply  284  by  121 

2.  Multiply  376  by  121. 


3.  Multiply  4740  by  12-^. 

4.  Multiply  70424  by  \2^. 


72.    To  multiply  by  331 

1.  Multiply  675  by  33^. 

Analysis. — Annexing  two  ciphers  to  the  mul-  operation. 
tiplicand,  multiplies  it  by  100 :  but  the  multiplier  3)67500 
is  but  one-third  of  1 00  :  lienee. 

Annex  two  ciphers  and  divide  the  result  by  3. 


22500 


EXAMPLES. 


1.  Multiply  889626  by  331. 

2.  Multipl'y  740362  by  331 


3.  Multiply  5337756  by  33f 

4.  MultipK  2221086  by  33^. 


68.  When  the  divisor  is  I,  what,  is  the  quotient!  When  the  divisor 
is  equal  to  the  dividend,-  what  is  the  quotient  I  When  the  divisor  is  less 
than  the  dividend,  how  does  the  quotient  compare  with  1  I  When  the  di- 
visor is  greater  than  the  dividend,  how  does  the  quotient  compare  with  I  ! 

09.   If  a  product  l)o  divided  by  one  of  the  lactors.  what  i.s  the  quotient ! 


70 


OONTK ACTION i?    IN    MULTU'LIOATION. 


73.   To  multiply  by  125. 

1,  Multiply  375  by  125. 

Analysis. — Annexing  three  ciphers  to  the  mul-  operation. 
tiplicand,  miiltipiie.s  it  by  1000:  but  125  i^  but  8)375000 
one-€i<rhth  of  one  thousand  :  hence, 


Annex  three  ciphers  and  divide  the  result  by  8. 


46875 


EXAIVIPLES. 


1.  Multiply  29632  by  125. 

2.  Multiply  8796704  by.l25. 


3.  Multiply  970406  by  125. 

4.  Multiply  704294  by  125. 


74.   By  reversing  the  last  four  processes,  we  have  the  four 
following  rules  : 

1.  To  divide  any  number  by  25  ; 

Multiply  the  number  by  A,  and  divide  (he  product  by  100. 

2.  To  divide  any  number  by  12^. 

Multiply  the  nmnher  by  8,  and  divide  the  product  by  100. 

3.  To  divide  any  number  by  33^  : 

Multiply  the  number  by  3,  and  divide  the  product  by  100. 

4.  To  divide  any  number  by  125  : 

Multipy  by  8,  and  divide  the  product  by  1000. 


EXAMPLES. 


1.  Divide  3175  by  25. 

2.  Divide  106725  bv  25. 

3.  Divide  21S7600  by  25. 

4.  Divide  2426225  by  25. 

5.  Divide  1762405  by  25. 

6.  Divide  4075  by  12^. 

7.  Divide  3550  bV  12^. 

8.  Divide  592621  by  121 


9. 
10. 
11. 
12. 
13. 
H. 
15. 
16. 


Divide  880300  by  12 J. 

Divide  22500  by'  331. 

Divide 

Divide 

Divide 

Divide 

Divide  3007875  by  125 

Divide  6758625  by  125. 


654200  bv  331 
7925200  bv  33J, 
4036200  bV  331 
93750  by  125. 


70.  What  is  the  rule  for  multiplying  by  25  1 

71.  What  is  the  rule  for  multiplying  by  12^  ? 

72.  What  is  the  rule  for  multiplying  by  33^  ? 
TO.   What  is  the  rule  for  multiplying  by  1251 


0>N"!RA(TriONS    IN    DIVISION.  71 


CONTRACTIONS  IN  DIVISION. 

75.  Contractions  in  Division  are  short  nnethods  of  finding 
the  quotient,  when  the  divisors  are  composite  numbers. 

CASE  I. 

76.   When  the  divisor  is  a  cow^posite  number. 

1.  Let  it  be  required  to  divide  1407  dollars  equally  among 
21  men.     Here  the  factors  of  the  divisor  are  7  and  3. 

Anai.vsis. — Let  the   1407   dollars 

be  first  divided  into  7  equal   piles.  operation. 

Each   pile  will  contain   201   dollars.  7)1407 

Lei  each  pile  be  now  divided   into  3  -^^  ^^^  quotient 

equal  parts.     Lack  part  will  contain ^ 

67  dollars,  and  the  number  of  parts  67   quotient  sought 
will  be  Jil  :  hence  the  foilowing 

Rule. — Divide  the  dividend  by  one  of  the  factors  of  the 
divisor ;  then  divide  the  quotient,  thus  arising,  by  a  second 
factor,  and  so  on,  till  every  factor  has  been  used  as  a  divisor  : 
the  last  qaoiient  will  be  the  answer. 

EXAMPLES. 

Divide  the  following  numbers  by  the  factors  ; 
1.   1260  by  12  =  3x4.  |  6.  55728  by  4  x9  x4=144. 


2.  18576  by  48  =  4x12. 

3.  9576  by  72  =  9x8. 

4.  19296  by  96-12x8. 


6.  92880  by  2  X  2  X  3  X  2  X  2. 

7.  57888  by  4x2x2x2. 

8.  154368  by  3x2x2. 


Note. — It  often  happens  that  there  are  remainders  after  some 
of  tlie  divisions.     How  are  we  to  find  the  true  remainder  ? 


74. — 1.  What  is  the  rule  for  dividing  by  25  ? 
2    What  is  the  rule  for  dividing  by  12^^? 

3.  What  is  the  rule  for  dividing  by  33^  ? 

4.  What  is  the  rule  for  dividing  by  125  ? 

7"^.  What  are  contractions  in  division  ?  What  is  a  compoiiite  num- 
ber' 

7ft.  W^hat  is  the  rule  for  division  when  the  divisor  is  a  composite 
number  ^  ' 


72  UONTRACTIONS 

77.  Let  it  be  required  to  divid©  751  grapes  into  16  equal 
parts. 

4X4=:16  ^  4)187  .  .  .  .3  first  remainder. 
^         46  ....  3x4=.12 


1 5  true  rem.     Ans.  i&j^. 
Note. — The  factors  of  the  divisor  16,  are  4  and  4. 

Analysis. — If  751  grapes  be  divided  by  4,  there  will  be  i87 
bunches,  each  containing  4  grapes,  and  3  grapes  over.  The  unit 
of  187  is  07ie  bunch  ;  that  is,  a  unit  4  times  as  great  as  1  grape. 

If  we  divide  187  bunches  by  4,  we  shall  have  46  piles,  each 
containing  4  bunches,  and  3  bunches  over  :  here,  again,  the  unit 
of  the  quotient  is  4  times  as  great  as  the  unit  of  the  dividend. 

If,  now  we  wish  to  find  tlie  number  of  gra])es  not  included  in 
the  46  piles,  we  have  3  bunches  with  4  grapes  in  a  bunch,  and 
3  grapes  besides  :  hence,  4x3  =  12  grapes  ,  and  adding  3 
grapes,  we  have  a  remainder,  15  grapes  ;  therefore,  to  find  the 
remainder,  in  units  of  the  given  dividend  : 

I.  Multiply  the  last  retnainder  by  the  last  divisor  hut  one, 
and  add  in  the  preceding  remainder  : 

II.  Multiply  this  result  by  the  next  preceding  divisoT", 
and  add  in  the  remainder,  and  so  on,  till  you  reach  the 
unit  of  the  dividend. 

EXAMPLES. 

1.  Let  it  be  required  to  divide  43720  by  45. 
r  3)43720 
45=3x5x3  \  5)r4o73    .l=ilstrem.     1x5  +  3  =  8; 
I     3)2914    .  3  =  2d  rem.     Sx2>-\-\^2b 
^  971     .  l=:3d  rem.  25  true  rem. 

Divide  the  following  numbers  by  the  factors,  for  the  divisors  : 


2.  956789  by  7x8  =  56. 

3.  4870029  by  8x9  =  72. 

4.  674201  by  10x11  =  110. 
6.  445767  by  12x12  =  144. 


6.  1913578  b>  7x2x3  =  42 

7.  146187  by"3x5x7  =  105 

8.  26964  by5x2xll=llC 

9.  93696  by3x  7x11  =  231 


T    Give  the  rule  for  the  remainder 


IN    DIVISION.  /3 

VASK    II. 

78.    When  the  divisor  is  10,   100,  1000,   ^c. 

Analysis. — Since  any  number  is  made  up  of  units,  tens,  hun- 
dreds &c.  (Art.  28),  the  number  of  tcTis  in  any  dividend  will 
denote  how  many  times  it  contains  1  ten,  and  the  units  will  be  the 
renuiinder.  The  hundreds  will  denote  how  many  times  the  divi- 
dend contains  1  hundred,  and  the  tens  and  unils  will  be  the  remain- 
der ;  and  similarly  when  the  divisor  is  1000,  10000.  &c.  ;  hence, 

Cut  off  from  the  right  hand  as  mani)  figures  as  there  are  ciphers  in 
the  divisor — the  figures  at  the  left  will  be  the  quotient,  and  those  at 
he  right,  the  renminder. 

EXAMPLES. 


1.  Divide  49763  by  lO. 

2.  Divide-7641200  by  100. 


3.  Divide  496321  by  1000. 

4.  Divide  64978  by  10000. 


79.  Whe7i  there  arc  ciphers  on  the  right  of  the  divisor, 

I.  Let  it  be  required  to  divide  673889  by  700. 
Analysis. — We  may  regard  the  operation. 

divisor  as  a  composite  number,  of     7100)673189 

which   the  factors   are  7   and   100.  T^         - 

We  first  divide  by  100  by  .^trikinij  -^^  '  '  ^  remains. 

off  the  89,  and  then  find   thai  7  is  _l2?.  ^^"^  remain. 

contained  in  the  remaining  tigures,  Ans.  961^^. 

96   times,  with  a  remainder  of  1  : 

this  remainder  we  multiply  by  100,  and  then  add  89,  forming  the 

true  remainder  189  ■:  to  the  quotient  96  we  annex  189  divided  by 

700,  for  the  entire  quotient  :  hence,  the  following 

Rule. — I.  Cut  off  the  ciphers  by  a  line,  and  cut  off  the 
same  number  of  figures  from  the  right  of  tJie  divide?id. 

II.  Divide  the  remaining  figures  of  the  dividend  by  the 
remaining  figures  of  the  divisor,  and  annex  to  the  remciin' 
der,  if  there  be  one,  the  figures  cut  off  from  the  dividend  : 
this  will  form  the  true  remainder. 

EXAMPLES. 

1.  Divide  8749632  by  37000. 

78.  How  do  you  divide  when  the  divisor  is  1  with  ciphers  annexed  ■ 
•Give  the  reason  of  the  rule  ! 

79.  How  do  you  divide  when  there  are  ciphers  on  the  light  of  the 
divisor  !     How  do  you  iorm  I  he  true  remahider '! 


74  AP  PI -I  CATION'S. 


a7|000)8749|632(236 
74 
134  Ans.  23611 1?  2 

111 


239 

222 

"17 

Divide  the  following  numbers 

2.  986327  by  210000. 

3.  870000  by  6000. 

4.  30599503  by  400700. 


5.  5714364900  by  36500 

6.  18490700  by  73000. 

7.  70807149  by  31500. 


APrLICATIONS. 

80.  Abstractly,  the  object  of  division  ia  to  find  from  two 
given  numbers  a  third,  which,  multiplied  by  the  first,  will 
produce  the  second.      Practically,  it  has  three  objects  : 

1.  Knowing  the  number  of  things  and  their  entire  cost,  to 
find  the  price  of  a  single  thing  : 

2.  Knowing  the  entire  cost  of  a  number  of  things  and  the 
price  of  a  single  thing,  to  find  the  number  of  things  : 

3.  To  divide  any  number  of  things  intb  a  given  immbei  of 
equal  parts. 

For  these  cases,  we  havp  from  the  previous  principles 
(page  57),  the  fijllowing 

RULES. 

I.  Divide  the  entire  cost  by  the  rium^ier  of  the  things  : 
the  quotient  will  be  the  price  of  a  single  thing. 

II.  Divide  the  entire  cost  by  the  ])rice  of  a  single  thing  : 
the  quotient  will  he  the  nutnher  of  things. 

III.  Divide  the  whole  number  of  things  by  the  nwmher  of 
parts  into  which  they  are  to  be  divided  :  the  quotient  will 
be  the  number  i?i  each  fart. 

QUESTIONS    INVOLVING    THE    PREVIOUS    RULES. 

1.  Mr.  Jones  died,  leaving  an  estate  worth  4500  dollars,  to 
be  divided  equally  between  3  daughters  and  2  sons  :  what 
wag  the  share  of  each  ? 


80   What  is  the  object  of  division,  abstractly  1   How  many  objects  has 
It,  practically  !   Nau)e  the  three  objects.  Give  the  rules  for  the  three  ca.se8 


AFPLIUATIOMS.  76 

2.  What  number  must  be  multiplied  by  124  to  produce 
40796? 

3.  The  sum  of  19125  dollars  is  to  be  distributed  equally 
among  a  certain  number  oi' men,  each  to  receive  425  dollars  : 
hoAv  many  men  are  to  receive  the  money  ? 

4.  A  merchant  has  5100  pounds  of  tea,  and  wishes  to  pack 
it  in  60  chests  :  how  much  must  he  put  in  each  chest  ? 

5.  The  product  ol'  two  numbers  is  51679680,  and  one  of 
the  factors  is  615  :  what  is  the  other  factor  1 

6.  Bourrht  156  barrels  of  flour  for  1092  dollars,  and  sold 
the  same  for  9  dollars  per  barrel :  how  much  did  1  gain '? 

7.  Mr.  James  has  14  calves  worth  4  dollars  each,  40  sheep 
worth  3  dollars  each  ;  he  gives  them  all  for  a  horse  worth 
150  dollars  :  does  he  make  or  lose  by  the  bargain  ? 

8.  Mr.  VYilsou  sells  4  tons  of  hay  at  12  dollars  per  ton.. 
80  bushels  of  wheat  at  1  dollar  per  bushel,  and  takes  in 
payment  a  horse  worth  65  dollars,  a  wagon  worth  40  dollars, 
and  the  rest  in  cash  :  how  much  money  did  he  receive  ? 

9.  How  many  pounds  of  coflee,  worth  12  cents  a  pound, 
must  be  given  for  368  pounds  of  sugar,  worth  9  cents  a 
pound  ? 

10.  The  distance  around  the  earth  is  computed  to  be  about 
25000  miles  :  how  long  would  it  take  a  man  to  travel  that 
distance,  supposing  him  to  travel  at  the  rate  of  35  miles  a 
day  ? 

11.  If  600  barrels  of  flour  cost  4800  dollars,  what  will 
2172  barrels  cost '? 

12.  If  the  remainder  is  17,  the  quotient  610,  and  the  divi- 
dend 45767,  what  is  the  divisor  1 

13.  The  salary  of  the  President  of  the  United  States  la 
25000  dollars  a  year  :  how  much  can  he  spend  daily  and 
feave  of  his  salary  4925  dollars  at  the  end  of  the  year  ? 

14.  A  farmer  purchased  a  farm  for  which  he  paid  18050 
dollars.  He  sold  50  acres  for  60  dollars  an  acre,  and  the  re- 
mainder stood  him  in  50  dollars  an  acre  :  how  much  land 
did  he  purchase  ? 

15.  There  are  31173  verses  in  the  Bible:  how  many 
verses  must  be  read  each  day,  that  it  may  be  read  through 
in  a  year] 

16.  A  farmer  wishes  to  exchange  250  bushels  of  oats  at 
42  cents  a  bushel,  for  flour  at  7  dollars  per  barrel  ;  how  many 
barrels  will  he  receive  ? 


70  AlTLlliATlONS 

17.  The  owner  of  an  estate  9o1q  <i40  acres  of  land  and  had 
S12  acres  left  :  how  .many  acres  h.-id  he  at  first? 

18.  Mr.  James  bought  of  Mr.  Johnson  two  farms,  one  con 
taining  250  acres,  for  which  he  paid  85  dollars  per  acre  ;  the 
second  containing  175  acres,  for  which  he  paid  70  dollars  an 
acre  ;  he  then  sold  them  botli  for  75  dollars  an  acre  :  did  he 
make  or  lose,  and  how  much  'I 

19.  A  farmer  has  279  dollars  with  which  he  wishes  to  buy 
cows  at  25  dollars,  sheep  at  4  dollars,  and  pigs  at  2  dollars 
apiece,  of  each  an  equal  number  :  how  many  can  he  buy  of 
each  sort  l 

20.  The  sum  of  two  numbers  is  3475,  and  the  smaller  is 
1162  :   what  is  the  greater? 

21.  The  diflerence  between  1m  o  numbers,  1475,  and  the 
greater  number  is  5760  :  what  is  the  smaller? 

22.  If  the  product  of  two  numbers  is  346712,  and  one  of 
the  factors  is  76  :  what  is  the  other  factor] 

23.  li  the  quotient  is  482,  and  the  dividend  135442  :  what 
is  the  divisor  ? 

24.  A  gentleman  bought  a  house  for  two  thousand  twenty- 
five  dollars,  and  furnished  it  for  seven  hundred  and  six  dol- 
lars ;  he  paid  at  one  time  one  thousand  ,and  ten  dollars,  and 
at  another  time  twelve  hundred  and  seven  dollars  :  how  much 
remained  unpaid  ? 

25  At  a  certain  election  the  wliole  number  of  votes  cast 
for  two  opposing  candidates  was  12672  :  the  successful  can- 
didate received  316  majoiity  :  how  many  votes  did  each  re- 
ceive ? 

26.  Mr.  Place  purchased  15  cows  ;  he  sold  9  of  them  for 
35  dollars  apiece,  and  the  remainder  for  32  dollars  apiece, 
when  he  found  that  he  had  lost  1 23  dollars  :  how  much  did 
he  pay  apiece  for  the  cows  ? 

27.  Mr.  Gill,  a  drover,  purchased  36  head  of  cattle  at  64 
dollars  a  head,  and  88  sheep  at  5  dollars  a  head  ;  he  sold  the 
cattle  at  one-quarter  advance  and  the  sheep  at  one-fifth  ad- 
vance :  how  much  did  he  receive  for  both  lots? 

28.  Mr.  Nelson  supplied  his  farm  with  4  yoke  of  oxen  at 
93  dollars  a  yoke  ;  4  plows  at  11  dollars  apiece  ;  8  horses  at 
97  dollars  each  ;  and  agrees  to  pay  for  them  in  wheat  at 
1  dollar  and  a  half  per  bushel  :  how  many  bushels  n.ust  he 
give  ? 


A.1'PL1CAT1UNIS.  77 

29.  If  a  man's  salary  is  600  (ioil  irs  a-year  and  his  expenses 
425  dollars,  how  many  years  will  elapse  hefbrc  he  will  bo 
worth  10000  dollars,  if  he  is  worth  2500  dollars  at  the  pre- 
Beut  time  ? 

30.  jHow  long  can  125  men  subsist  on  an  amount  of  food 
that  will  last  1  man  4500  days  ? 

31.  A  speculator  bought  512  barrels  of  flour  for  3584  dol- 
lars and  sold  the  same  for  4608  dollars :  how  much  did  he 
gain  per  barrel  1 

32.  A  merchant  bought  a  hogshead  of  molasses  containing 
96  gallons  at  35  cents  per  gallon  ;  but  26  gallons  leaked  out, 
and  he  sold  the  remainder  at  50  cents  per  gallon  :  did  he 
gain  or  lose,  and  how  much  ? 

33.  Two  persons  counting  their  money,  together  they  had 
342  dollars  ;'  but  one  had  28  dollars  more  than  the  other  : 
how  many  had  each  i 

34.  Mrs.  Louisa  Wilsie  has  3  houses,  valued  at  12530  dol- 
lars, 1 1324  dollars,  and  9875  dollars  :  also  a  farm  worth  6720 
dollars.  She  has  a  daughter  and  2  sons.  To  the  daughter 
she  gives  one-third  the  value  of  the  houses  and  one-iburth  the 
value  of  the  farm,  and  then  divides  the  remainder  equally 
among  the  boys  :  how.  much  did  each  receive  ? 

35.  A  person  having  a  salary  of  1500  dollars,  saves  at  the 
end  of  the  year  405  dollars  :  what  were  his  average  daily 
expenses,  allowing  365  days  to  the  year  i 

36.  Mr.  Bailey  has  7  calves  worth  4  dollars  aiDAOce, 
y  sheep  worth  3  dollars  apiece,  and  a  fine  horse  worth  175 
dollars.  He  exchanges  them  for  a  yoke  of  oxen  worth  125 
dollars  and  a  colt  worth  65  dollars,  and  takes  the  balance  in 
hogs  at  8  dollars  apiece  :  ho\^  many  does  he  take  ? 

37.  Mr.  Snooks,  the  tailor,  bought  of  Mr.  tSquire,  the  mer- 
chant, 4  pieces  of  cloth  ;  the  first  and  second  pieces  each 
measured  45  yards,  the  third  47  yards,  and  the  fourth  53 
yards  ;  for  the  whole  he  paid  760  dollars  :  what  did  he  pay 
for  35  yards  ? 

38.  Mr.  Jones  has  a  farm  of  250   acres,  worth  125  dollar 
pur   acre,  and   offers    to   exchange  with   Mr.   Cushiiig,   whose 
farm  contains  185  acres,  provided  Mr.  Oushing  will  pay  him 
20150    dollars    difference:   what    was    Mr.    Cushing's    farm 
valued  at  per  acre  1 


78  Ai'PLic/.ii(.)Xc;. 

39.  The  volcano  in  the  island  of  Bourbon,  in  1796,  threw 
.wt  45000000  cubic  feet  of  lava  :  how  long  would  it  take  25 
carts  to  carry  it  off,  ii"  each  cart  carried  12  loads  a  day,  and 
40  cubic  feet  at  each  load  ? 

40.  The  income  of  the  Bishop  of  Durham,  in  England,  is 
292  dollars  a  day  ;  how  many  clergymen  would  this  support 

n  a  salary  qf  730  dollars  per  annum  ? 

41.  The  diameter  of  the  earth  is  7912  miles,  and  the  diame- 
ter of  the  sun  112  times  as  great  ;  what  is  the  diameter  oi'the 
sun. 

42.  By  the  census  of  1850,  the  whole  population  of  the 
United  iStates  was  23191876  ;  the  number  oi  births  for  the 
previous  year  was  629444  and  the  number  of  deaths  324394  : 
supposing  the  births  to  be  the  only  source  of  increase,  what 
was  the  population  at  the  beginning  of  the  previous  year  '^ 

43.  Mr.  Sparks  bought  a  third  part  of  neighbor  Spend- 
thrift's farm  for  2750  dollars.  Mr.  Spendthrift  then  sold  half 
the  remainder  at  an  advance  of  250  dollars,  and  then  Mr. 
Sparks  bought  what  was  left  at  a  further  advance  of  250 
dollars  :  how  much  money  did  Mr.  Sparks  pay  Mr.  Spend- 
thrift, and  what  did  he  get  for  his  whole  farm  ? 

44.  George  Wilson  bought  24  barrels  of  pork  at  14  dollars 
a  barrel  ;  one-fourth  of  it  proved  damaged,  and  he  sold  it  at 
half  price,  and  the  remainder  he  sold  at  an  advance  of  3  dol- 
lars a  barrel :  did  he  make  or  lose  by  the  operation,  and  how 
much  ? 

45.  A  miller  bought  320  bushels  of  wheat  for  576  dollars, 
and  sold  256  bushels  for  480  dollars  :  what  did  the  remain- 
der cost  him  per  bushel  ? 

46.  A  merchant  bought  117 -yards  of  cloth  for  702  dollars, 
and  sold  76  yards  of  it  at  the  same  price  for  which  he  bought 
it ;  what  did  the  cloth  sold  amount  to  ? 

47.  If  46  acres  of  land  produce  2484  bushels  of  corn  •  how 
many  bushels  will  120  acres  produce  ] 

48.  Mr.  J.  Williams  goes  into  business  wdth  a  capital  of 
25000  dollars  ;  in  the  first  year  he  gains  2000  ;  in  the  second 
year  3500  dollars  ;  in  the  third  year  4000  dollars  ;  he  then 
invests  the  whole  in  a  cargo  of  tea  and  doubles  his  money  ; 
he  then  took  out  his  original  capital  and  divided  the  residue 
equally  between  his  5  children  :  wdiat  was  the  portion  of 
each  1 


dNHED    STATES    MONEY.  79 


UNITED    STATES    MONEY. 

81.  Numbers  are  collections  of  units  of  the  same  kind, 
[n  Ibrinmg  these  collections,  we  first  collect  the  lowest  or  pri- 
mary units,  until  we  reach  a  certain  number ;  we  then 
chanfre  the  unit  and  make  a  second  collection,  and  after 
reaching  a  certain  number  we  again  change  the  unit,  and  so  on 

In  abstract  numbers,  we  first  collect  the  units  1  till  we 
reach  ten  ;  we  then  change  the  unit  to  1  ten  and  collect  till 
we  reach  10  ;  we  then  change  the  unit  to  100,  and  so  on. 

A  Scale  expresses  the  relations  between  the  orders  of  units, 
in  any  number.  There  are  two  kinds  of  scales,  uniform  and 
varying.  In  the  abstract  numbers,  the  scale  is  uniform,  the 
units  of  the  scale  being  10,  at  every  point. 

82.  United  States  money  is  the  currency  established  by 
Congress,  A,  D.  1786.  The  names  or  denominations  of  its 
units  are,  Eagles,  Dollars,  Dimes,  Cents,  and  Mills. 

The  coins  of  the  United  States  are  of  gold,  silver,  and  cop- 
per, and  are  of  the  following  denomuiations  : 

1.  Gold  :  Eagle,  half-eagle,  three-dollars,  quarter-eagle, 
dollar. 

2,  Silver:  Dollar,  half-dollar,  quarter-dollar,  dime,  half- 
dime,  and  three-cent  piece. 

3    Copper:  Cent,  half-cent. 

TABLE. 


10  Mills     make 

1   Cent,  marked  ct. 

10  Cents 

. 

1  Dime,     •     -     d. 

10  Dimes 

-     - 

1    Dollar,    -     .     $. 

10  Dollars 

-    - 

1   Eagle,    -     -  E. 

* 

Mills. 

Cents. 

Dimes.         Dollars. 

Eaglei 

10 

=    1 

100 

=r     10 

=    1 

1000 

=r     100 

=    10            =1 

10000 

=    1000 

=    100          =    10 

=   1 

81.  What  are  numbers  ^  How  are  numbers  formed  !  How  are  sim- 
ple numbers  formed  \  What  is  the  scale  \  What  is  the  primary  unit 
in  simple  numbers  1 


80  UNITED    STATES   MONEY. 

83.  It  is  seen,  from  the  above  table,  that  in  United  States 
money,  the  primary  unit  is  1  mill  ;  the  units  of  the  scale,  in 
passing  from  mills  to  cents,  are  10.  The  second  unit  is  1 
cent,  and  the  units  of  the  scale  in  passing  to  dimes,  are  10. 
The  third  unit  is  1  dime,  and  the  units  of  the  scale  in  passing 
to  dollars,  are  10.  The  fourth  unit  is  1  dollar,  and  the  uniti 
of  the  scale  in  passing  to  eagles,  are  10.  This  scale  is  th« 
sam^e  as  in  simple  nu^nbers  ;  therefore, 

The  units  of  United  States  money  may  be  added,  sub 
traded,  mvltipUed,  and  divided,  by  the  mme  rules  that 
have  already  been  given  far  simp)le  numbers. 

i  NUMEK  \TION  TABLE. 


^ 

i 

N 

g 

s 

09 

O 

03 

CO 

■t.^ 

1 

^-1 

•  <<-i 

O 

^   °   «i 

09 

^    m  ii 

JOB 

e 

=    G    C 

^&^6% 

5  7,  is  read  5  cents  and  7  mills,  or  o7  mills. 
16  4,     -     -   16  cents  and  4  mills,  or  164  mills. 
6  2,  1  2  0,     -     -  62  dollars  12  cents  and  no  mills. 
2  7,623,     -     -  27  dollars  62  cents  and  3  mills. 
4  0,  0  4  1,     -     -  40  dollars  4  cenis  and  1  mill. 

The  comma,  or  separatrix,  is  generally  used  to  separate  the 
cents  from  the  dollars.  Thus  $67,256  is  read  67  dollars  25 
cents  and  6  mills.  Cents  occupy  the  two  first  places  on  the 
right  of  the  comma,  and  mills  the  third. 

United  States  money  is  read  in  dollars,  cenis  and  mills. 


82.  What  is  United  States  money  ?  What  aie  the  names  of  its 
units  1  What  are  the  coins  of  the  United  States  1  Which  gold  1 
W^hich  silver  1     Which  copper  1 

83.  in  United  States  money  what  is  the  primary  unit "?  W^hat  is  the 
scale  in  passing  from  one  denomination  to  another  1  How  does  this 
compare  with  the  scale  in  simple  numbers  ?  What  then  follows  \ 
What  is  used  to  separate  dollars  from  cents  1  How  is  United  States 
money  read  ! 

84.  X^'hat  is  reduction  1  How  many  kinds  of  reduction  are  there? 
Name  them.  How  may  cents  be  changed  into  mill«  "  How  mav  dol. 
lars  be  changed  into  cents'?     How  into  mills  ^ 


UNITED   STATES    MONEY.  81 


REDUCTnN  OF  UNITED  STATES  MONEY. 

84.  Reduction  of  United  States  Money  is  changing  the 
unit  from  one  denomination  to  that  of  another,  without  altering 
the  value  of  the  number.     It  is  divided  into  tvro  parts  : 

]  St  To  reduce  from  a  greater  unit  to  a  less,  as  from  dol- 
lars to  cents. 

2d.  To  reduce  from  a  less  unit  to  a  greater,  as  from  mills 
to  dollars. 

85.   To  reduce  from  a  greater  unit  to  a  less. 

From  the  table  it  appears, 

1st  Tlmt  cents  may  be  changed  into  mills  by  annexing 
one  cipher. 

2d  That  dollars  may  be  changed  into  cents  by  annexifig 
two  ciphers,  and  into  mills  by  annexing  three  ciphers. 

3d.  That  eagles  may  be  cJianged  into  dollars  by  annexing 
one  cipher. 

The  reason  of  these  rules  is  evident,  since  10  mills  make  a 
cent,  100  cents  a  dollar,  and  1000  mills  a  dollar,  and  10 
dollars  1  eagle. 

EXAMPLES. 

1.  Reduce  25  eagles,  14  dollars,  85  cents  and  6  mills  to 
the  denomination  of  mills. 

OPERATION. 

25  eagles  =  250  dollars, 
add    14  dollars, 

264  dollars= 26400  cents, 
add         -         -  85  cents, 

26485  cents  =  264850  mills, 

add 6  mills, 

Ans.  264856  mills, 

2.  In  3  dollars  60  cents  and  5  miUs,  how  many  mills  ? 

3  dollar8=300  cents, 

60  cents  to  be  added, 
360  =  3600  mills,  to  which  add  the  5  mills. 


82  REDUCTION    OF 

3.  In  37  dollars  37  cents  8  mills,  how  many  mills  ? 

4.  In  375  dollars  99  cents  9  mills,  how  many  mills? 

5.  How  many  mills  in  67  cents  1 
6  How  many  mills  in  i^54  ? 

7.  How  many  cents  in  |>125? 

8.  In  ^400,  how  many  cents'?     How  many  mills? 

9.  In  $375,  how  many  cents  ?     How  many  mills? 

10.  How  many  mills  in  ^4  ?    In  |;6  ?    In  $10,14  cents? 

11.  How  many  mills  in  ^40,36  cents  8  mills'? 

12.  How  many  mills  in  •1>71,45  cents  3  mills? 

86.    To  reduce  from  a  less  unit  to  a  greater. 

1.  How  many  dollars,  cents  and  mills  in  26417  mills'^ 
ANALVfiis. — We  hist  divide  the  mills  by  10,  operation 

giving  2641  cents  and   7  Jiiills  over;  we  then         10)2641j7 
divide  the  cents  by  100,  giving  26  dollars,  and       100^26141 
41  cents  over:  hence,  the  answer  is  26  dollars  '      ' 

41  cents  and  7  mills:  therefore,  Ans.  ^26,417 

I.  To  reduce  mills  to  cents  :  cut  off  the  right  hand  figure 

II.  To  reduce  cents  to  dollars  :  cut  off  the  two  right  hand 
figures :   and, 

III.  To  reduce  mills  to   dollars  :  cut  off  the   three  right 
hand  figures. 

EXAMPLES. 

1.  How  many  dollars  cents  and  mills  are  there  in  67897 
mills  ? 

2.  Set  down  104  dollars  69  cents  and  8  mills. 

3.  Set  down  4096  dollars  4  cents  and  2  mills. 

4.  Set  down  100  dollars  1  cent  and  1  mill. 

5.  Write  down  4  dollars  and  6  mills. 

6.  Write  down  109  dollars  and  1  mill. 

7.  Write  down  65  cents  and  2  mills. 

8.  Write  down  2  mills. 

9.  Reduce  1607  mills,  to  dollars  cents  and  mills. 

10.  lleduce  170464  mills,  to  dollars  cents  and  mills. 

11.  Reduce  6674416  mills,  to  dollars  cents  and  mills. 

12.  Reduce  94780900  mills,  to  dollars  cents  and  mills. 

13.  Reduce  74164210  mills,  to  dollars  cents  and  mills. 

86.   How  do  you  change  mills  into  cents  !     How  do  you  change  cent* 
into  dollars'     How  do  you  change  mills  lo  dollars  \ 


rNTTFD    STATES 

87.  One  nii/iiber  is  said  to  be  an 
when  it  is  contained  in  that  other  an  exact  nuiliuui  Ultimes. 
Thus  ;  50  cents,  25  cents,  &c.,  are  aliquot  parts  of  a  dollar  : 
so  also  2  months,  3  months,  4  months  and  6  months  are  ali- 
quot parts  of  a  year.  The  parts  of  a  dollar  are  sometimes 
expressed  fractionally,  as  in  the  following 


TABLE  OF  ALIQUOT  PARTS 

$;  =100  cents. 

^  of  a  dollar^:  50  cents. 
J  of  a  dollar  ==33^  cents. 
^  of  a  dollars:  25  cents. 
i  of  a.  dollars   20  cents. 


•i-    of  a  dollar :=:  121  cents. 
y\j  of  a  dollars    10  cents. 


Tfi 

2^0  of  a  dollar = 


'4 

5  cents. 


^   of  a  cent    =      5  mills. 


ADDITION  OF  UNITED  STATES  MONEY. 


and 


3^  cents  for  6 


1 .  Charles  gives  9^   cents   for   a   top 
quills  :  how  much  do  they  all  cost  him  ? 

2.  John  gives  $1,37^  lor  a  pair  of  shoes,  25  cents  for  a 
penknife,  and  12^  cents  for  a  pencil  :  how  much  does  he  pay 
ibr  all  I 


Analysis. — We  observe  that  half  a  cent  is  equal 
to  5  mills.  We  then  place  the  mills,  cents  and  dol- 
lars in  separate  columns.  We  then  add  as  in  simple 
numbers. 


3.  James  gives  50  cents  for  a  dozen  oranges, 
12^  cents  for  a  dozen  apples,  and  30  cents  for 
a  pound  of  raisins  :  how  much  for  all  ? 


OPERATION. 

$1,3/5 
,25 
,125 

§1750 

OPZRATION. 

$0,50 
,125 
,30 

$0,925 


88.  Hence,  for  the  addition   of  United   States  money,  we 
have  the  following 

Rule. — I.   Set  down  the  numbers  so  that  units  (^  the 
same  value  shall  fall  in  the  same  column. 

87.  What  is  an  aliquot  part  1     How  many  cents  in  a  dollar  I     hi  half 
a  dollnr'?     h)  a  lliird  of  a  t'ollar  ■?     in  a  li)uith  uf  a  tln.'l.'ir'! 


84  AP PLICATIONS    IN 

II.   Add  up  the  several  coluinns  as  in   simple  nwm^ers^ 

and  place  the  separating  point  in  the  sum  directly  undet 
that  in  the  columns. 

Pkoof. — The  same  as  in  simple  numbers 

EXAMPLES. 

1.  Add  $67,214,    $10,049,    |6,041,  $0,271,  togethoi. 

(1.)                                    (2.)  (3.) 

$  cts.  m.                             $  cts.  m.  $  cts.  m. 

67,214             59,316  81,053 

10,049             87,425  67.412 

6,041             48,872  95,376 

,   0,271             56.708  87.064 


$83,575  $330,905 

APPLICATIONS. 

1.  A  grocer  purchased  a  box  of  candles  for  6  dollars 
89  cents  :  a  box  of  cheese  for  25  dollars  4  cents  and  3  mills  ; 
a  keg  of  raisins  for  1  dollar  12^  cents,  (or  12  cents  and  5 
mills  ;)  and  a  cask  of  wine  lor  40  dollars  37  cents  8  mills  : 
what  did  the  whole  cost  him  % 

2.  A  farmer  purchased  a  cow  for  which  he  paid  30  dollars 
and  4  mills;  a  horse  for  which  he  paid  104  dollars  60  cents 
and  1  mill  ;  a  wagon  for  which  he  paid  ^b  dollars  and 
9  mills  :  how  much  did  the  whole  cost  ? 

3.  Mr.  Jones  sold  farmer  Sykes  6  chests  of  tea  for  §75,641  ; 
9  yards  of  broadcloth  for  $27,41  ;  a  plow  for  -^9,75  ;  and  a 
harness  for  $19,674  :  what  was  the  amount  of  the  bill? 

4.  A  grocer  sold  Mrs.  WiUiams  18  hams  ibr  $26,497  ;  a  bag 
of  coffee  for  $17,419  ;  a  chest  of  tea  for  $27,047  ;  and  a 
firkin  of  butter  for   $28,147  :  what  was  the  amount  of  her 

biin 

5.  A  father  bought  a  suit  of  clothes  for  each  of  his  four 
boys;  the  suit  of  the  eldest  cost  $15,167  ;  of  the  second, 
$13,407  ;  of  the  third,  12,75  ;  and  of  the  youngest,  $11,047  ; 
how  much  did  he  pay  in  all  ? 

88.  How  do  you  set  down  the  numbers  for  addition  1  How  do  you 
(idd  up  the  columns  1  How  do  you  place  the  sf-puratinif  point  '  How 
it!  you  j-Tove  ;wldiliuii  \ 


1 


UNITKI)   STATES    MOJi^FY.  85 

6.  A  father  has  six  children  ;  to  the  first  two  he  gives 
each  $375,416  ;  to  each  of  the  second  two,  'i5287,DO  ;  to  each 
of  the  third  two,  $259,004  :  how  much  did  he  give  to  them 

air^ 

7.  A  man  is  indebted  to  A,  $630,49  ;  to  B,  $25  ;  to  C, 
87^  cents;  to  D,  4  mills  :  how  much  does  he  owe  i 

8.  Bought  1  gallon  of  molasses  at  28  cents  per  gallon  ;  a 
Balf  pound  of  tea  for  78  cents  ;  a  piece  of  flannel  for  12  dol- 
lars  6  cents  and  3  mills  ;  a  plow  for   8  dollars,   1  cent  and 

1  mill ;  and  a^pair  of  shoes  lor  1  dollar  and  20  cents  :  what 
did  the  whole  cost  ? 

9.  Bought  6  pounds  of  coffee  for  1  dollar  12|  cents  ;  a 
wash-tub  for  75  cents  6  mills ;  a  tray  for  26  cents  9  mills  ;  a 
broom  9^-'  Zl  cents  ;  a  box  of  soap  for  2  dollars  65  cents 
7  miU* ,  -c  cheese  for  2  dollars  87-^  cents ;  what  is  the  whole 
amount " 

10.  "'•^^a'^  :s  the  entire  cost  of  the  following  articles,  viz.  : 

2  gallons  f"^  molasses,  57  cents;  half  a  pound  of  tea,  37^ 
cents  ;  2  yards  of  broadcloth,  |3,37^  cents;  8  yards  of  flan- 
nel, $9,875  ;  two  skeins  of  silk,  12^  cents,  and  4  sticks  of 
twist,  8A  cents  ? 

SUBTRACTION  OF  UNITED  STATES  MONEY. 

1 .  John  gives  ^  cents  lor  a  pencil,  and  5  cents  for  a  top 
how  much  more  does  he  give  for  the  pencil  than  top? 

2.  A  man  buys  a  cow  for  $26,37,  and  a  calf  for  $4,50  : 
how  much  more  does  he  pay  for  the  cow  than  calf? 

OPERATION. 

Note. — We  set  down  the  numbers  as  in  addition,      $26.37 
and  then  subtract  them  as  in  simple  numlers.  4,50 

$21,87 

89.  Hence,  for  subtraction  of  United  States  money,  we 
have  the  following 

Rule. — I.  Write  the  leas  number  under  the  greater  so  that 
units  of  the  same  value  shall  fall  in  the  same  column. 

89.  How  do  you  set  down  the  numbers  for  subtraction  1  How  do 
you  subtract  them  1  Where  do  you  place  the  separating  pjiut  iu  the 
ivuiiiijilcr ^     How  do  you  provr  i^uStr.irtinu  ! 


86  BUHTRACTK^N    (»K 

II.  Subtract  as  in  simijle  nvmbcrs^  and  place  the  separating 
point  in  the  remainder  directly  under  that  in  the  columns. 

Proof. — Tlie  same  as  in  simple  numbers. 

EXAMPLES. 

(1.)  (2.) 

From              $204,679  From  $8976,400 

Take                  98,714  Take  610.098 

Remainder   "|l05!^965  Remainder  $8366^2 

(3.)  (4.)                          (5.) 

1620,000  $327,001  $2349 

19,021  2,090                       29,33 

$600;979  $324,911  $2319,67 

6.  What  is  the  difference  between  $6  and  1  mill?  Between 
$9,75  and  8  mills?  Between  75  cents  and  6  mills?  Between 
$87,354  and  9  mills  ? 

7.  From  $107,003  take  $0,479. 

8.  From  $875,043  take  $704,987. 

9.  From  $904,273  take  $859,896. 

APPLICATIONS. 

1.  A  man's  income  is  $3000  a  year  ;  he  spends  $187,50  : 
how  much  does  he  lay  up  ? 

2.  A  man  purchased  a  yoke  of  oxen  for  $78,  and  a  cow  for 
$26,003  :  how  much  more  did  he  pay  ibr  the  oxen  than  for 
the  cow  1 

3.  A  man  buys  a  horse  for  $97,50,  and  gives  a  huxidred 
dollai  bill :  how  much  ought  he  to  receive  back  1 

4.  How  much  must  be  added  to  $60,039  to  make  the  sum 
$1005,40? 

5.  A  man  sold  his  house  for  $3005,  this  sum  being  $98,039 
more  than  he  gave  for  it  :  what. did  it  cost  him? 

6.  A  man  bought  a  pair  of  oxen  for  $100,  and  sold  thcrn 
again  for  $75,37^  :  did  he  make  or  lose  by  the  bargain,  and 
how  much  ? 

7.  A  man  starts  on  a  journey  with  $100  ;  he  spends 
$67,57  :  how  much  has  he  left? 

8.  How  much  iiiuFt  you  add  to  $40,173  U.  make  $100? 


UNITED    STATES    MONET.  87 

9.  A  man  purchased  a  pair  of  horses  for  f$450>  but  finding 
)ne  of  them  injured,  the  seller  agreed  to  deduct  $106,325  : 
what  had  he  to  pay  ? 

10.  A  farmer  had  a  horse  worth  $147,49,  and  traded  him 
for  a  colt  worth  but  $35,048  :  how  much  should  he  receive 
in  money  ? 

11.  My  house  is  worth  $8975,034  ;  my  barn  $695,879: 
what  is  the  difierence  of  their  values'? 

12.  What  is  the  diilerence  between  nine  hundred  and  sixty- 
nine  dollars  eighty  cents  and  1  mill,  and  thirty-six  dollars 
ninety-nine  cents  and  9  mills  ? 

MULTIPLICATION  OF  UNITED  STATES  MONEY. 

1.  John  gives  3  cents  apiece  for  6  oranges  :  how  much  do 
they  cost  him  ? 

2.  John  ,buys  6  pairs  of  stockings,  for  which  he  pays  25 
cents  a  pair  :  how  much  do  they  cost  him  ? 

3.  A  farmer  sells  8  sheep  for  $1,25  each:  how  much  does 
he  receive  ibr  them  ? 

OPERATION. 

Analysis. — We  multiply  the  cost  of  one  sheep  by  $1,25 

the  number  of  sheep,  and  the  product  is  the  entire  8 

^^^'  $10,00 

90.  Hence,  for  the  multiplication  of  United  States  money 
by  by  an  abstract  number,  we  have  the  following 

Rule. — I.  Write  the  money  for  the  multiplicand,  and  the 
abstract  number  for  (he  tnaltijAier. 

II.  Multiply  as  in  simple  numbers,  and  the  product  ivill 
be  the  answer  in  the  lowest  deyiomination  of  the  midtv- 
plica  nd.  * 

III.  Reduce  the  prroduct  to  dollars,  cents  and  mills. 
Proof. — Same  as  in  simple  numbers. 

EXAMPLES. 

1.  Multiply  385  dollars,  28  cents  and  2  mills,  by  8. 
operation.  (2.) 

$385,282  $475,87 

8  9 


Product    $3082,250  Product  $42S2,83 


^3  MUT.TIPTJCATIOX    OF 

3.  What  will  55   yards  of  cloth  come  to   at   37  cents  pei 
yard  ? 

4.  What  will  300  bushels  of  wheat  come  to  at  $1,25  per* 
bushel ? 

5.  What  will   85  pounds  of  tea  come  to  at   I   dollar  Z7^ 
cents  per  pound  ? 

6.  What  will  a  firkin  of  butter  containing  90  pounds  ccme 
to  at  25-J-  cents  per  pound  ? 

7.  What  is   the  cost  of  a  cask  of  wine  containing  29  gal- 
lons, at  2  dollars  and  75  cents  per  gallon  ? 

8.  A  bale  of  cloth  contains   95    pieces,  costing  40  dollars 
37J  cents  each  :  what  is  the  cost  of  the  whole  bale  1 

9.  W4iat  is  the  cost  of  300  hats  at  3  dollars  and  25  centfl 
apiece  ? 

10.  What  is  the  cost  of  9704  oranges  at  3^  cents  apiece  1 

OPERATION. 

Note. — We  know  that  the  product  of  two  nura-  9704 

bars  contains  tiie  same  number  of  uniis,  whichever  034 

be  used   as  the    multiplier  (Art.  48).     Hence,   we  4 8 52~ 

may  multiply  9704   by  3^  if  we  assign  the  proper        „        ^ 
unit  (1  cent)  to  the  product. 

8339,64 

11.  What  will   be  the  cost  of  356  sheep  at  3^  dollars  a 
head  I 

12.  What  will  be  the   cost  of  47  barrels  of  apples  at  1| 
dollars  per  barrel  ? 

13.  What  is  the  cost  of  a  box  of  oranges  containing  450, 
at  2-^  cents  apiece  ? 

14.  What  is  the   cost  of  307   yards  of  linen  at  68^  cents 
per  yard  ? 

15.  What  will  be  the  co^t  of  65  bushels  of  oats  at  331  cents 
a  bushel  ? 

Analysis. — If  the   price  were  1  dollar  a  bushel,     operation, 
the  cost  would   be   as  many   dollars   as   there   are       3)65,000 
bushels.     But  the  cost  is  33i  cents=i  of  a  dollar:        g2^r666^ 
hence,  the  cost  will  be  as   many  dollars  as  3  is  con-  '      ^ 

tained   times  in  65  =  21   dollars,  and  2  dollars  over,  which  is  re- 


90.  How  do  you  multiply  United  States  money  1  What  will  he  the 
denomination  of  the  product^  How  will  you  then  reduce  it  to  dollars 
'jnd  rents!?     How  do  yoji  prove  nmHiiili<'Mion1 


UWiniD   STATE©   MOiN-KY.  89 

duced  to  cents  by  adding  two  ciphers,  and  to  mills  by  adding 
three ;  then,  dividing  the  cents  and  mills  by  3,  we  have  the  entire 
cost :  hence, 

91.  To  find  the  cost,  when  the  price  is  an  aliquot  part  of 
a  dollar. 

Take  such  a  part  of  the  mimher  which  denotes  the  commo- 
dity,  as  the  price  is  of  1  dollar. 

EXAMPLES. 

1.  What  would  be  the  cost  of  345  pounds  of  tea  at  50 
cents  a  pound  ? 

2.  What  would  675  bushels  of  apples  cost  at  25  cents  a 
bushel  ? 

3.  If  1  pound  of  butter  cost  12^  cents,  what  will  4  firkins 
cost,  each  weighing  56  pounds'? 

4.  At  20  cents  a  yard,  what  will  42  yards  of  cloth  cost  ? 

5.  At  33^  cents  a  gallon,  what  will  136  gallons  of  mo- 
lasses cost  1 

OPERATION. 

6.  What  will  1276  yds.  4)$1276  cost  at  1  dollar  a  yard. 
of  cloth  cost  at  $1,25  a  319  cost  at  25  cts.  a  yard, 
y^^^^  •                                             $1595  cost  at  $1,25  a  yard. 

7.  What  would  be  the  cost  of  318  hats  at  ^1,121  apiece  ? 

8.  What  will  2479  bushels  of  wheat  come  to  at  $1,50 
a  bushel  1 

9.  At  $1,33^  a  foot,  what  will  it  cost  to  dig  a  well  78  feet 
deep? 

1 0.  What  will  be  the  cost  of  936  feet  of  lumber  at  3 
dollars  a  hundred  ? 

Analysis. — At  3  dollars  a  foot  the  cost  would  be  operation. 
936X3  =  2808  dollars;  but  as  3  dollars  is  the  price  936 

of  100  feet,  it  follows  that  2808  dollars  is  100  times  3 

ilie  cost  of  the  lumber  :  therefore,  if  we  divide  ^^^^  ^^ 
'.808  dollars  by  1 00  {which  we  do  by  cutting  oflT  two  5^^o,uo 
t»f  the  right  hand  figures  (Art.  73),  we  shall  obtain  the  cost. 

Note. — Had  the  price  been  so  much  per  thousand,  we  should 
have  divided  by  1000,  or  sut  off  three  of  the  right  hand  figures* 
ttence. 


91.  How  do  you  fmd  tlu?  coft  of  several  thitigs  when  the  price  is  an 
aliquot  part  of  u  diAlarl 

4 


tfO'  MULllPLICATiON    Of 

92.  To  find  the  cost  of  articles  sold  by  the  100  or  1000  : 
Multiply  the  quantity  by  the  jrrice  ;  and  if  the  price  be 
by  the  100,  cut  off  t-wo  figures  on  the  right  hand  of  the 
product  ;  if  by  the  1000,  cut  off  three,  and  the  remaining 
figures  will  be  the  answer  in  the  same  denmnination  as  Ih/e 
price,  which  if  cents  or  imlls,  may  be  reduced  to  dollars. 

EXAMPLES. 

1.  What  will  4280  bricks  cost  at  $5  per  1000? 

2.  What  will  2673  feet  of  timber  cost  at  $2,25  per  100  ? 

3.  What  will  be  the  cost  of  576  feet  of  boards  at  $10,62 
per  1000? 

4.  What  is  the' value  of  1200  feet  of  lathing  at  7  dollars 


T  1000? 

5.  David  Trusty, 

Bought  of  Peter  Bigtree, 

2462  feet  of  boards 

at  $7,       per  1000. 

4520     " 

"      9,50 

600     "     scantling 

"    11,37 

960     "     timber 

«    15, 

1464     "     lathing 

,75  per  100 

1012     "     plank 

"      1,25 

Received  Payment, 

Peter  Bigtree, 

6.  What  is  the  cost  of  1684  pounds  of  hay  at  $10,50  pei 
ton! 

Analysis.  —  Since  there    are  operation. 

2000/6.   in   a   ton,   the    cost   of         2)10,50 
1000/6  will  be  half  as  much  as  —^^       -^^  ^^  ^^^q^ 

for   1   ton:   viz.    $5.2o,  or   525  iaqa 

cents.      Multiply   this    by    the  ^"^^ 

number  of  pounds   (1684),   and       $8,84100  Ans. 
cut   off    three    places    from    the 
right,  in  addition  to  the  two  places  before  cut  off  for  cents  :  henoo, 

93.  To  find  the  cost  of  articles  sold  by  the  ton  : 
Multiply   one-half  the  price  of  a  ton  by  the  number  of 
pounds,  and  cut  off  three  figures  from  the  right  hand  of 
the  pi'oduct.     The  remaining  figures  will  be  the  ansic&r  in 
the  same  denomination  as  the  price  of  a  ton. 


92.   Huw  do  yuu  find  the  co^t  uf  articles  siAdhy  tiio  100  or  KMK)  7 


UNriKD   STATES   MONEY.  91 

EXAMPLES. 

1.  "WTiat  will  3426  pounds  of  plaster  cost  at  $3,4^  per  ton  1 

2.  What  will  be   the    cost  of"  the    transportation  of"  6742 
pounds  of  iron  firom  Buflalo  to  New  York,  at  $7  per  ton  ? 

J.  What  will  be  the   cost   of  840  pounds   of  hay  at  $9,50 
per  ton'.'  at  $12  ?  at  $15,84?  at  $10,361  at  $18,75? 

DIVISION  OF  UNITED  STATES  MONEY. 

94.  To  divide  a  number  expressed  in  dollars,  cents  or  mills, 
into  any  number  of  equal  parts. 

Rule. — I.  Reduce  the  dividend  to  cents  or  mills,  if  necessary, 
II.  Divide  as  in  simple  numbers,  and  the  quotient  will  be  the 

answer  in  ike  lowest  denomination  of  the  dividend:  this  may 

be  reduced  to  dollars^  cents,  and  mills. 

Proof. — Same  as  in  division  of  simple  numbers. 

Note. — The  sign  -f  is   added  in  the  examples,  to  show  that 
there  is  a  remainder,  and  that  the  division  may  be  continued. 

EXAMPLES. 

1.  Divide  $4,624  by  4  ;  also,  $87,256  by  5. 

OPERATION.  OPERATION. 

4)$4,624  5)$87,256 


$1,156  $17,451^ 

2.  Divide  $37  by  8. 

Analvsis. — In  this  example  we  first  reduce  the  oPERATinN. 

$37  to  mills   by  annexing  three  ciphers.     The  quo-  8)$37,000 

tient  will  then  be  mills,  and  can  be  reduced  to  dol-  ^~X625 

lars  and  cents,  as  before.  ' 

3.  Divide  856,16  by  16. 

4.  Divide  S495,704  by  129. 

5.  Divide  $12  into  200  equal  parts. 

6.  Divide  $4  00  into  600  equal  parts. 
7    Divide  $857  into  51   equal  parts. 

8.  Divide  $6578,95  into  157  equal  parts. 

93.  How  do  you  find  the  cost  of  articles  sold  by  the  ton  1 

94.  What  is  the  rule  for  division  of  United  States  money  !  Haw  do 
you  prove  division  1  How  do  you  indi<:aie  ihr.t  the  divisjion  maybe 
continue*]? 


y^  DlVIbJON    OF 

95.  The  quantity,  and  the  cost  of  a  quantity  given,  to  find 
the  price  of  unity  (Art.  80). 

Divide  the  cost  hy  ike  quantity, 

9.  Bought  9  pounds  of  tea  for  $5,85  ;  what  was  the  price 
per  pound  ? 

10.  Paid|;29,68  for  14  barrels  of  apples:  what  was  the 
price  per  barrel  % 

11.  If  27  bushels  of  potatoes  cost  $10,125,  what  is  the 
price  of  a  bushel  % 

12.  If  a  man  receive  $29,25  for  a  rriunth's  work,  how 
much  is  that  a  day,  allowing  26  working  days  to  the  month  1 

13.  A  produce  dealer  bought  3  barrels  of  eggs,  each  con- 
taining 150  dozens,  for  which  he  paid  $63  :  how  much  did 
he  pay  a  dozen  ? 

14.  A  man  bought  a  piece  of  cloth  containing  72  yards, 
for  which  he  paid  $252  :  what  did  he  pay  per  yard  \ 

15.  If  $600  be  equally  divided  among  26  persons,  what 
will  be  each  one's  share  ? 

16.  Divide  $18000  in  40  equal  parts  :  what  is  the  value  of 
each  part  % 

17.  Divide  $3769,25  into  50  equal  parts  :  what  is  one 
part? 

18.  A  farmer  purchased  a  farm  containing  725  acres,  for 
which  he  paid  $18306,25  :  what  did  it  cost  him  per  acre  % 

19.  A  merchant  buys  15  bales  of  goods  at  auction,  for 
which  he  pays  $1000  :   what  do  they  cost  him  per  bale  ? 

20.  A  drover  pays  $1250  for  500  sheep  ;  what  shall  he 
sell  them  lor  apiece,  that  he  may  neither  make  nor  lose  by 
the  bargain  % 

21.  The  dairy  of  a  farmer  produces  $600,  and  he  has  25 
cows  :  how  much  does  he  make  by  each  cow  ? 

22.  A  farmer  receives  $840  for  the  wool  of  1400  sheep : 
how  much  does  each  sheep  produce  him  ? 

23.  A  merchant  buys  a  piece  of  goods  containing  105 
yards,  for  which  he  pays  $262,50  ;  he  wishes  to  sell  it  so  as 
to  make  $52,50  :  how  much  must  he  ask  per  yard  ? 

96.  When  the  price  of  unity  and  the  cost  of  a  quantity  are 
given,  to  find  the  quantity  (Art.  80). 

Note. — The  divipor  and  dividend  must  both  be  redueca  to  tlie 
lowest  iiiul  niinicd  in  cither  bc.'btc  (livi<liiit:. 


UNrrED    STATES    MONEY.  U3 

Divide  the  cost  hy  the  price, 

24.  If  I  pay  $4,50  a  ton  for  coal,  how  much  can  I  buy 
for  ^67,501 

25.  At   $7  a  barrel,  how  much  flour  can  be  bought  foi 
$178,50? 

26.  How  many  pounds  of  tea  can  be  bought  for  $6,75,  at 
75  cents  a  pound  ? 

27.  What   number  of  barrels  of  apples  can  be  bouglit  for 
$47,50,  at  $2,37^  a  barrel  ? 

28.  At  44  cents  a  bushel,  how  many  bushels  of  oats   can 
be  bought  for  $14,30  ? 

29,' At  34  cents  a  bushel,  how  many  barrels  of  apples  can 
I  buy  for  $13,60,  allowhig  2J  bushels  to  the  barrel  ? 

30.  If  1   acre  of    land  cost   $28,75,   how  much  can    be 
bought  for  83220  ? 

31.  Paid  $40,50  for  a  pile  of  wood,  at  the  rate  of  $3,37^ 
a  cord,  how  much  Avas  there  in  the  pile  ? 

32.  How  many  sheep  can  be  bought  for  $132,  at  $1,37^  a 
head  ? 

33.  At   $4,25   a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $68  ? 

34.  At   $1,12^  a  day,  how  long  would  it  take  a  person  to 
earn  $157,50  ? 

APPLICATIONS    IN    THE    FOUH    PRECEDING    RULES. 

Note. — See  and  repeat  Rule — page  53 :  also  the  three  rules — 
jage  74. 

1.  If  1  yard  of  cloth  costs  3^  dollars,  what  will  8  yards  costi 

2.  If  1  ton  of  hay  costs  S14i,  what  will  9  tons  cost? 

3.  If  1  calf  costs  $4^,  what  will  12  calves  cost  ? 

4.  Mr.  Jones  bought  250  bushels  of  oats,  for  which  he  paid 
^156,25  :  how  much  did  they  cost  him  a  bushel  '\ 

5    If  12   tons  of  hay  cost   150   dollars,  what   does    1  tou 
cost  ?  8  tons  ?   50  tons  ? 

6.  If  9   dozen  oi'  spelling  books  cost  $7,875,  what  will  ) 
dozen  cost  1   6  dozen  ?  8  dozen  ? 

7.  If  75  bushels  of  wheat  cost  $131,25,  how  much  will   1 
bushel  cost?  8  bushels?   120  bushels  \ 

8.  If  320  pounds  oi"  coflee  cost  $44,80  cents,  how  much 
will  1  pound  001.4  ?     What  will  575  pounds  co?t  '• 


04  APPLICATIONB    IN 

9.  Mr.  James  B.  Smith  bought  9  barrels  of  sugar,  each 
weighing  216  pounds,  for  which  he  paid  $]  16,64  :  how  much 
did  he  pay  a  poutid  ? 

10.  if  40  tons  of  hay  cost  ^580,  how  much  is  that  per 
ton  ?     What  M'ould  70  tons  cost  at  the  same  rate  ? 

11.  If  Mr.  Wilson  has  $120  to  buy  his  winter  wood,  and 
wood  is  $4  a  cordrhow  many  cords  can  he  buy  ] 

12.  At  6  dollars  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  24  dollars  ?     How  many  for  $36  ? 

13.  A  farmer  sold  a  yoke  of  oxen  for  $80,75  ;  6  com's  for 
$29  each ;  30  sheep  at  $2,60  a  head  ;  and  3  colts,  one  for 
$25,  the  othei  two  lor  $30  apiece  :  what  did  he  receive  for 
the  whole  lot  ? 

14.  A  merchant  buys  6  bales  of  goods,  each  containing  20 
pieces  of  broadcloth,  and  each  piece  of  broadcloth  contained 
29  yards  ;  the  whole  cost  him  $15660  :  how  many  yards  of 
cloth  did  he  purchase,  and  how  much  did  it  cost  him  pc 
yard  ? 

15.  A  person  sells  3  cows  at  $25  each  ;  and  a  yoke  of 
oxen  for  $65  ;  he  agrees  to  take  in  payment  60  sheep  :  how 
much  do  his  sheep  cost  him  per  head  ? 

16.  A  man  dies  leaving  an  estate  of  $33000  to  be  equally 
divided  among  his  4  children,  after  his  wife  shall  have  taken 
her  third.  What  was  the  wife's  portion,  and  what  the  part 
of  each  child  1 

1 7.  A  person  settling  with  his  butcher,  finds  that  he  is 
charged  with  126  pounds  of  beef  at  9  cents  per  pound  ;  85 
pounds  of  veal  at  6  cents  per  pound  ;  6  pairs  of  fowls  at  37 
cents  a  pair  ;  and  three  hams  at  $1,50  each  :  how  much 
does  he  owe  him  ? 

18.  A  farmer  agrees  to  furnish  a  merchant  40  bushels  of 
rye  at  62  cents  per  bushel,  and  to  take  his  pay  in  coffee  at 
16  cents  per  pound  :  how  much  coffee  will  he  receive  ? 

19.  A  farmer  has  6  ten-acre  lots,  in  each  of  which  he  pas- 
tures 6  cows  ;  each  cow  produces  112  pounds  of  butter,  for 
which  he  receives  18^  cents  per  pound  ;  the  expenses  of 
each  cow  are  5  dollars  and  a  half:  how  much  does  he  make 
by  his  dairy  1 

20.  Bought  a  farm  of  W.  N.  Smith  for  2345  dollars,  a 
span  of  horses  for  375  dollars,  6  cows  at  36  dollars  each  ;  I 
paid  him  520  dollars  in  cash,  and  a  village  lot  worth  1500 
•llll^<.r?;  :   }u»w  mrniv  ddllurs  vMnn.in  ii»!;»fii(l  'i 


UHriTED    STATES   MONEY.  96 

BILLS    OF    PARCELS. 

(21.)  New  York,  May  1st,  1854. 

Mr.  James  Spendthrift, 

Bovffht  of  Benj.  Saveall, 
16  pounds  of  tea  at  85  cents  per  pound     -     -     - 
27  pounds  of  cofiee  at  15^  cents  per  pound     -     - 
15  yards  of  linen  at  66  cents  per  yard  -     .     -     - 

i 

Received  payment,  Benj,  Saveall. 


(22.)  Albany,  June  2d,  1854. 

Mr.  Jacob  Johns, 

Bought  of  Gideon  Gould, 
36  pounds  of  sugar  at  9i  cents  per  pound       -     - 
3  hogsheads  of  molasses,  63  galls,  each,  at  27  ) 

cents  a  gallon j 

6  casks  of  rice,  285  pounds  each,  at  5  cents  per  ) 

pound J 

2  chests  of  tea,  86  pounds  each,  at  96  cents  per  ) 

pound J 

Total  cost,     $ 
Received  payment,         For  Gideon  Gould, 

Charles  Clark* 


(23.)  Hartford,  November  21st,  1854. 

Gideon  Jvnes^ 

Bought  of  Jacob  Thrifty, 
69  chests  of  tea  at  $55,65  per  chest  -     -     - 
126  bags  of  coilee,  100  pounds  each,  at   12J 

cents  per  pound 

167  boxes  of  raisins  at  S2,75  per  box       -     - 
800  bags  of  almonds  at  Si 8,50  per  bag  -     - 
9004  barrels  of  shad  at  $7,50  per  barrel    -     - 
60  barrels  of  oil,  32  gallons  each,    at  $1,08 
per  gallon 

Amount,     $ 
Received  the  above  in  full,  Jacob  Thrifty. 


96  DENOMINATE    NUMBEliS. 


DENOMINATE    NUMBERS. 

97.  A  Simple  number  is  a  unit  or  a  collection  of  units. 
The  unit  may  be  either  abstract  or  denominate. 

98.  A  Denominate  number  is  a  collection  of  denominate 
units  :  thus,  3  yards  is  a  denominate  number,  in  which  the 
unit  is  1  yard. 

99.  Numbers  which  have  the  same  unit,  are  of  the  same 
denomination  :  and  numbers  having  different  units,  are  of 
different  denominations.  If  two  or  more  denominate  num- 
bers, having  different  units,  are  connected  together,  forming  a 
single  number,  such  is  called  a  compound  denominate  number. 

100.  There  are  eight  different  units  in  Arithmetic  :  1st. 
The  abstract  unit  :  2d.  The  unit  of  currency  :  3d.  The  unit 
of  length  :  4th.  The  unit  of  surface  :  5th.  The  cubic  unit  or 
unit  ot  volume  :  6th.  The  unit  of  weight  :  7th.  The  unit  of 
time  :  8th.  The  unit  of  circular  measure. 

ENGLISH    MONEY. 

101.  The  units  or  denominations  of  English  money  are 
guineas,  pounds,  shillings,  pence,  arid  farthings. 

TABLE. 

4  farthings  marked  far,  make  1  penny,     marked       d, 
12  pence  -  -         -         -  1  shilling,         -  s. 

1  pound,  or  sovereign,  j£ 
1  guinea. 

s.  £ 

=  1 

=  20  =1 

Notes. — 1.  The  primary  unit  in  English  money  is  1  farthing, 
'""he  number  of  units  in  the  scale,  in  passing  from  farthings  to 

97.  What  is  a  simple  number  ^ 

98.  What  is  a  denominate  number  ( 

99.  When  are  numbers  of  the  same  denomination  ?  When  of  differ- 
^\  t  denominations  1  If  several  numbers  having  different  units  are  con- 
nected together,  what  is  the  number  called  ! 

100.  How  many  units  are  there  in  Arithmetic  ]     Name  them. 


20  shillings 

- 

21  shillings 

•                     m 

far. 

d. 

4 

=      1 

48 

=  12 

960 

=  240 

DKNOMINAl'E    NUMBKRS.  97 

pence,  is  4;  in  passing  from  pence  to  shillings,  12  ;  in  passing 
from  shillings  to  pounds,  20. 

2.  Farthings  are  generally  expressed  in  fractions  of  a  penny. 
Thus,  Ifar.  —  id.;  2far.=^id.;  '6  far.  =  Id. 

3.  By  reading  the  second  table  from  right  to  left,  we  can  see 
the  value  of  any  unit  expressed   in  each  of  the  lower  denomina 

ions.      Thus,    ld.=Afar.  ;  ls.  =  nd.=4Sfar. ;    £1=205.^2406/. 
=  960/rtr. 

REDUCTION  OF  DENOMINATE  NUMBERS. 

102.  Reduction  is  changing  the  unit  of  a  number,  without 
altering  its  value. 

1.  How  many  pence  are  there  in  2^.  6d.  ? 

Analysis. — Since  there  are  12  pence  in  1  sliilling,  there  are 
twice  12,  or  24  pence  in  2  shillings  :  add  the  6  pence  :  therefore, 
in  25.  6d.  there  are  30  pence. 

2.  How  many  pence  in  4  shiUings  1  In  45.  8c?.  ?  In  5s. 
6d.  ?     In  35.  8d.  1     In  65.  Td.  ? 

3.  How  many  shillings  in  £21     In  £3  85.,  how  many  ? 

4.  How  many  pence  in  £1  ?  How  many  shillings  in 
£2  85.  1     How  many  in  £3  75.  ? 

5.  How  many  shillings  are  there  in  48  pence  ? 

Analysis. — Since  there  are  12  pence  in  1  shilling,  there  are  as 
many  shillings  in  48  pence,  as  12  is  contained  times  in  48,  which 
is  4  :  Iherelore,  there  are  4  shillings  in  48  pence. 

6.  How  many  pounds  in  40  shillings  ?     In  GO  ?     In  80  ? 

103.  From  the  above  analyses  we  see,  that  reduction  of 
denominate  numbers  is  divided  into  two  parts  : 

1st.  To  change  the  unit  of  a  number  from  a  higher  detio- 
mi  nation  to  a  lower. 

2d.  To  change  the  unit  of  a  number  from  a  lower  denomi- 
nation to  a  higher. 

101.  What  are  the  denominations  of  English  money  \ 

Notes.  1. — What  is  the  primary  unit  in  Engli.sh  money  1  Name  the 
scales. 

2. — How  are  farthings  generally  expressed  ? 

3. — How  is  the  second  table  read  1     What  does  it  show  1 

102.  What  is  Reduction  ! 

103.  Into  how  n)any  parts  is  reduction  divided  1     What  are  tUeyl 

7 


98  REDUCTION    OF 


PRINCIPLES    AND    EXAMPLES. 

104.  To  reduce  from  a  idyher  to  a  lower  unit. 

1,  Reduce  £27  65.  Sd.  to  the  denomination  of  farthings. 

Analysis. — Since  there  are  20  sliillings  in  operation. 

£\,'m  £'11  there  are  27  times  20  shillings,  £27  6^.  M.  2 far. 

or  540  shillings,  and  6  sliiliings  added,  make  20 

5465.     Since  12  pence  make  1   shilling,  we  — j-rr. — 

next  mnltiply  by  12,  and  then  add  M.  I.o  the  ^ \^^' 

product,  giving  6560  pence.     Since   4   far-  ^^ 

things  make  1  penny,  we  next  multiply  by  6560(/. 

4.  and    adi  2  farthings  to  the  product,  giv-  4 

ing  26242  farthings  lor  the  answer.  o^-  n  ■  o~  a 

26242  Ans, 

Note. — The  units  of  the  scale,  in  passing  from  pounds  to  shil. 
lings,  are  20;  in  passing  from  shillings  to  pence  they  are  12; 
and  in  passing  from  pence  to  farthings,  4. 

Hence,  to  reduce  from  a  higher  to  a  lower  unit,  we  have 
the  Ibllowing 

Rui>E. — Mnltiply  the  highest  denomination  hy  the  vnifs  of 
the  scale  which  connect  it  ivilh  the  next  lower.,  and  add  to  the 
product  the  vnita  of  that  denomination  :  proceed  in  the  same 
manner  ihrovyh  all  the  dcno)ninatiun.v^  till  the  unit  is  brought 
to  the  required  denomination. 

105.  To  reduce  from  a  lower  unit  to  a  higher. 
1.  Reduce  3138  farthings  to  pounds. 

OPERATION. 

Analysis.  —  Since     4    farthings  4)3138 

make  a  pennv,  we  first  divide  by  4.  . 

Since  12  pence  make  a  shilling,  we  i^'.^"  '^J^'^'  ^^"^• 

next  divide   by  12.     Since  20  shil-  2|0)6;5  -  -  Ad.  rem. 

linss  make  a  pound,  we  next  divide  3  -  -  -  5^?    rem 

by"'20,  and  find  that  3138/a/-.=£3  .„— -T-.rT,   4^    9  far 

Hence,  to  reduce  from  a  lower  to  a  higher  denomination, 
we  have  the  following 

Rule. — I.  Divide  the  given  number  by  the  units  of  the  scale 

104.  How  do  you  reduce  from  a  higher  to  a  lower  unit  1 

105.  How  do  you  reduce  from  a  lower  to  a  higher  unit  1  What  will 
be  the  unit  of  any  remainder  1     How  do  you  prove  reduction  '' 


UKNOMINATE   NUMBERS.  99 

which  connect  it  with  the  next  higher  denomination,  and  set 
down  the  remainder,  if  there  be  one, 

II.  Divide  the  quotient  thus  obtained  by  the  units  of  th( 
scale  ivhich  connect  it  with  the  next  higher  denornijiaiion,  and 
set  down  the  remainder. 

III.  Proceed  in  the  same  way  to  the  required  denomination^ 
and  the  last  quotient,  with  the  several  remainders  annexed^ 
will  be  the  answer. 

Note. —  Every  remainder  will  be  of  the  same  denomination  as 
its  dividend. 

Proof. — After  a  number  has  been  reduced  from  a  hif^ber 
denomination  to  a  lower,  by  the  first  rule,  let  it  be  reduced 
back  by  the  second  ;  and  alter  a  number  has  been  reduced 
from  a  lower  denomination  to  a  hipfher,  by  the  second  rule, 
let  it  be  reduced  back  by  the  first  rule.  If  the  work  is  right, 
the  results  will  agree. 

EXAMPLES. 

1.  Keduce  £15  7^.  6d.  to  pence. 

OPERATION.  PROOF. 

15  12)36'JQ 

—  ^FQ)30i7  .  .  .  6d.  rem. 
307  15       ...  7s.  rem. 

12 

3G90  Ans.  £15  75   6c/. 

2.  In  £31  8.S'.  9d.  ofar.,  how  many  farthings?  Also  proof 

3.  In  £87  14s.  8^(1.,  how  many  farthings'?     Also  proof. 

4.  In  £407  196'.  life/.,  how  many  farthings'?     Also  proof. 

5.  In  80  guineas,  how  many  pounds '? 

6.  In  1549/(/r  ,  how  many  pounds,  shillings  and  pence  ? 

7.  In  6169  pence,  how  many  pounds  ? 

LINEAR  MEASURE. 

106.  This  measure  is  used  to  measure  distances,  lengths, 
breadths,  heights  aud  depths,  &c 

106.  For  what  is  Linear  Measure  used  1  What  are  its  denominations  1 
Repeat  the  tahle.  What  is  a  fathom]  What  is  a  hand]  What  are 
the  units  of  the  scale. 


100 


KEDUCTIOiV   OF 


TABLE, 
12  inches  make 

3  feet       -  -  - 

51  yards  or  161  feet  - 
40  rods     -  .         -         - 

8  furlongs  or  320  rods  - 
3  miles    -         -         -         - 
691   statute  miles  (nearly)  or  ) 
60  geographical  miles,  ) 

360  degrees,         ... 
in.  ft.  yd. 

12  =   1 

36  =3  =1 

198  =    16^         =   5^ 

7920  =   660  =   220 

63360       =   5280       =    1760 


foot, 

yard, 

rod,  perch,  or  pole 

furlong,    - 

mile, 

league,     - 

degree  of  |    ^^ 


1 

1 
1 
1 
1 
1 
1 

the  equator, 

a  circum'nce  of  the  earth. 
rd. 


marked    Ji. 

yd, 

rd. 

fur. 

inl. 


[] 


fun 


mi. 


=   1 

=   40 
=   320 


Notes. 


A  fathom  is   a  length  of  six  feet. 


=    1 

=   8=1 
and  is  generally 


used  to  measure  the  depth  of  water. 

2.  A  band  in  4  inches,  used  to  measure  the  height  of  horses. 

3.  The  units  of  the  scale,  in  passing  from  inches  to  feet,  are  12  ; 
in  passing  from  feet  to  yards,  3  ;  from  yards  to  rods,  5-^ ;  from 
rods  to  furlongs,  40 ;  and  from  furlongs  to  miles,  8. 

1.  How  many  inches  in  5  feet?     In  10  feet  1     In  16  feet? 

2.  How  many  yards  in  36  hetl     In  54  feef?     In  96  ? 

3.  How  many  feet  in  144  inches'?     In  96  inches?     In  48  ? 

4.  How  many  furlongs  in  3  miles  1     In  6  miles  ?     In  8  ? 


EXAMPLES. 

1.  How  many  inches 
6rd.  iyd.  2ft.  '^in. 

in 

2.   In   1365  inches,  how 
many  rods? 

OPERATION. 

Grd.  4.1/d.  2ft.  9m. 

OPERATION. 

12)1365 

3)113  feet       9in. 

3 
34 
37  yards. 

3 

5^)37  yards    2ft. 
11)74 

6rd.  8halfyds.  =  42/d 

113  feet. 
12 

Ans.  Qrd.  Ayd.  2ft.  9in. 

1365  inches. 

DKNOMINATE    NUMBERS.  101 

Note. — When  we  reduce  rods  to  yards,  we  multiply  by  the 
scale  5^ ;  that  is,  we  take  6  rods  6  and  one-half  times.  When  we 
reduce  yards  to  rods,  we  divide  by  5^,  which  is  done  by  reducing 
the  dividend  and  divisor  to  halves  :  the  remainder  is  8  half-yards, 
equal  to  4  yards. 

3.  In  59mi.  Ifur,  38rc?.,  how  many  feet? 

4.  In  115188  rods,  how  many  miles'? 

5.  In  719mi.  16/-^.  Qyd.,  how  many  feet] 

6.  In  118°,  how  many  miles? 

7.  In  -54°  45mi.  Ifur.  20rd.  4:i/d.  2ft.  10m.,  how  many 
inches  ? 

8.  In  481401716  inches,  how  many  degrees,  &c.  ? 

CLOTH  MEASURE. 

107.  Cloth  measure  is  used  for  measuring  all  kinds  of 
cloth,  ribbons,  and  other  things  sold  by  the  yard. 


TABLE. 

2\  inches,  in 

I.     make 

1   nail,     marked 

na* 

4    nails 

- 

1    quarter 

of  a  yard, 

qr. 

3    quarters 

- 

1   Ell  Flemish," 

E.FL 

4    quarters 

- 

1   yard. 

- 

yd. 

5    quarters 

- 

1   Ell  Enghsh,      - 

E.E, 

in.              na» 

qr. 

E.FL 

yd. 

E.E. 

^         =  1 

9          =4 

=  1 

27          =12 

=  3 

r=   1 

36          =  16 

=  4 

—  ^i 

=  1 

45          =  20 

=  5 

—  ^ 

=  ii 

=  1 

Note. — The  units  of  the  scale,  in  this 
and  5. 

measure,  are  2^,  4,  3,  4 

1.  In  9  inches, 

how  many  nails  ? 

Plow  many  nails  in  1 

yard  ?      In  2  yard 

Is?     In  6? 

In  8  ? 

. 

2.  In  4  yards, 

how  many  quarters  ? 

How  many  quarters 

in  8  yards  ?     In  7  how  many  1 

3.  How  many  quarters  in 

12  nails  ? 

In  16  nails?     In  20 

nails?      In  36] 

In  40? 

^07.  For  what  is  cloth  measure  used  1    What  are  its  deuominatiuna  \ 
Rf'Ueat  the  table.     What  are  the  units  oi  the  scales  \ 


102 


KLpLO'ilON    OF 


liXAiMPLKS. 


1.  How  many  nails-  are 
there  in  35yd.  Sgr^Sna.  ] 

OPERATION. 

o5i/d,  3qr.  ona. 
4 

143  quarters. 
4 


2.  In  575  nails,  how 
many  yards  ? 

OPERATION. 

4)575 
4)  J  43  3na. 
35  dqr. 

Ana.  35yd.  3qr.  ona. 


575  nails. 

In  49  E.  E.,  how  many  nails? 
In  51  E.  Ft.,  2qr.  3na.,  how  many  nails? 
In  3278  naiis,  how  many  yards  ? 
In  340  nails,  how  many  Ells  Flemish] 
7.   In  4311  inches,  how  many  E.  E.  ? 

SQUARE  MEASURE. 

108.  Square  measure  is  used  in  measuring  land,  or  anything 
in  which  length  and  breadth  are  both  considered. 

1  Foot. 
A  square  is  a  figure  bounded  by  four  equal 
lines  at  right  angles  to  each  other.     Each 
line  is  called  a  side  of  the  square,     if  each 
side   be   one   foot,  the    figure   is  called   a    '"' 
square  foot. 

1  yard  =  3  feet 
If  the  sides  of  the  square  be  each  one 
yard,  the  square  is  called  a  square  yard. 
In  the  large  square  there  are  nine  small 
squares,  the  sides  of  which  are  each  one 
foot.  Therefore,  the  square  yard  contains 
9  square  leet.  " 

The  number  of  small  squares  that  is  contained  in  any  large 
square  is  always  equal  to  the  product  of  two  of  the  sides  of  I  he 
large  square.  As  in  the  figure,  3  x  3  =  9  square  feet.  The  number 
of  square  incites  contained  in  a  square  foot  is  equal  to  1 2  X  1 2  ~  1 44. 

108.  For  what  is  Square  Measure  used  1  What  is  a  square  1  If 
each  side  be  one  foot,  what  is  it  called  ?  If  each  side  be  a  yard,  what 
is  it  called  1  How  many  square  foct  does  the  square  yard  contain  ? 
How  IS  the  number  of  small  squares  contained  in  a  large  square  found] 
Repeat  tlie  table.     What  are  the  units  of  the  scale  1 


CO 

II 

DENOMINATE 


144  square  inches,  sq  in. 

9  square  feet 
30A  square  yards 
40  square  rods  or  perches 

4   roods     - 
540  acres     - 


fi- 

square  yard,  Sq  yd 

square  rod  or  perch,     P. 
rood,         -  R. 

acre,  -  A. 


1  square  mile. 


M. 


Sq.  in. 
144   ' 
1296 
39204 
1568160 
6272640 


Sq.ft. 

1 

9 

2721 

10890 


Sq.yd. 


=  1 

=  301 
=  1210 


=  43560   =  4840  = 


1 

40 
160 


R. 


=   1. 


Note. — The  units  of  the  scale  are  144,  9,  30-^,  40,  and  4.     106 

1.  How  many  square  inches  in  2  square  feet  ?  How  many 
square  feet  in  3  square  yards  ]     How  many  in  6  ?     In  8  ? 

2.  How  many  perches  in  1  rood  ?  In  3  roods  ?  How  many 
roods  in  4  acres  ?     In  8  ]     In  1 2  ? 

3.  How  many  perches  in  an  acre?  How  many  in  2  acres! 
How  many  square  yards  in  8 1  square  feet  ? 

SURVEYORS'  MEASURE. 

109.  The  Surveyor's  or  Gunter's  chain  is  generally  used  in 
Burveying  land.  It  is  4  poles  or  66  feet  in  length,  and  ia 
divided  into  100  links. 


TABLE. 


^tVo  ii^ches 


make         1  link,  marked   -         -  I. 

1  chain,     -         -         -  c. 

1  mile,       -         -         -  mi. 

16  square  rods  or  perches,  P. 

1   acre,        -         -         '  A. 

Note — 1.  Land  is  generally  estimated  in  square  miles,  acres, 
roods,  and  square  rods  or  perches. 

2,  The  unitss  of  the  scale  are  7j^g-,  4,  80,  1,  and  10. 


4  rods  or  66ft. 

80  chains    - 

1  square  chain  - 

10  square  chains 


109.  What  chain  is  used  in  land  surveyinof  1  What  is  its  length  ? 
How  is  it  divided  I  Repeat  the  table.  In  what  is  land  generally  esti- 
mated ?     What  are  the  units  of  the  scale  1 


104 


RKDUCTK^N    OF 


1.  How  many  rods  m  1  chain  ?     How  many  in  4  ?     In  5  f 

2.  How  many  chains  in  1  mile  ]    In  2  miles  1     In  3  ] 

3.  How  many  perches  in  1  square  chain  1     In  4  ?     In  6  ? 

4.  How   many   square  chains   in    2    acres  ?     How   many 
perches  in  3  acres  ?     In  5  ?     In  6  1 


EXAMPLES. 


1 .  How  many  perches  in 
32if.  25 A.  Sli.  19P  ? 

OPERATION. 

'S2M.  25 A,  3E.  19P. 
640 


20505    acres. 
4 


82023    roods. 
40 


2.   How     many     square 
miles,  &c.,  in  3280939jP.  ? 

OPERATION. 

40)3280939 

4)82023     i9P. 

640)20505       SB. 

32      25^1. 


Ans.  32M.  25 A.  ZJR.  19P. 
3280939  perches. 

3.  In  19^.  2Ii.  31  P.,  how  many  square  rods  ? 

4.  In  175  square  chains,  how  many  square  feet  ? 

5.  In  37456  square  inches,  how  many  square  feet  ? 

6.  In  14972  perches,  how  many  acres? 

7.  In  3674139  perches,  how  many  square  miles? 

8.  Mr.  Wilson's  farm  contains  104^1.  3B.  and  19P.  ;  he 
paid  for  it  at  the  rate  of  75  cents  a  perch  :  what  did  it  cost? 

9.  The  ibur  walls  oi'  a  rooni  are  each  25  feet  in  length  and 
9  feet  in  height  and  the  ceiling  is  25  feet  square  :  how  much 
will  it  cost  to  plaster  it  at  9  cents  a  square  yard  ? 

CUBIC  MEASURE. 

110.  Cubic  measure  is  used  for  measuring  stone,  timber 
earth,  and  such  other  things  as  have  the  three  dimensions, 
length,  breadth,  and  thickness. 

TABLE. 
1728  cubic  inches,  Cu.  in.  make   1   cubic  foot,  Cu.  ft. 

27  cubic  feet,  -         -         -  1   cubic  yard,  Cu.  yd 

40  feet  of  round  ^^  )  j  |on     -         -  T 

50  fieet  of  hewn  timber,  ) 

42  cubic  feet    -         -         -         1  ton  of  shipping,   T. 

16  cubic  feet    -  "  1   cm  d  foot,     -  C.  ft 

8  cord  feet  or  >        _  j^^^j  .         ^ 

125  «uhifi  feet,  \ 


II  ill 

III  III 

'^ 

DENOMINATE   NUMUEK8.  105 

Note. — 1.  A  cord  of  wood  is  a  pile  4  feet  wide,  4  feet  high, 
and  8  feel  loiig. 

,    2.  A  cord  foot  is  1  foot  in  length  of  the  pile  which  makes  a 
cord. 

3.  A  CUBE  is  a  figure  bounded  by  six  equal  squares,  called 
faces  ;  the  sides  of  the  squares  are  called  edges. 

4.  A  cubic  foot  is  a  cube,  each  of  whose  faces  is  a  square  foot; 
ts  edges  are  each  1  foot. 

5.  A  cubic  yard  is  a  cube,  each  of 
whose  edges  is  1  yard.  '^ 

6.  The  base  of  a  cube  is  the  face  >. 
on  which  it  stands.  If  the  edge  of  r* 
the  cube  is  one  yard,  it  will  contain  ii 
3X3  =  9  square  feet  j  therefore,  9  ^ 
cubic  feet  can  be  placed  on  the  base,  n 
and  hence,  if  the  figure  were  1  foot 
thick,  it  would  contain  9  cubic  feet;  3  feet=l  yard. 

if  it  were  2  feet  thick  it  would  contain  2  tiers  of  cubes,  or  18  cubic 
feet ;  if  it  were  3  feet  thick,  it  would  contain  27  cubic  feet ;  hence, 

The  contents  of  a  figure  of  this  form  are  found  by  miilti- 
l^lymg  the  length,  breadth,  and  thickness  together. 

7.  A  ton  of  round  timber^  when  square,  is  supposed  to  produce 
40  cubic  feet ;  hence,  one-fifth  is  lost  by  squaring. 

1.  In  1  cubic  foot,  how  many  cubic  inches  ?  How  many 
in  2  1     In  3  ? 

2.  In  1  cubic  yard,  how  many  cubic  feet  ?  How  many  in 
2  '?     In  41     In  6  ? 

3.  How  many  cord  feet  in  3  cords  of  wood  ?  In  5  ?     In  6  ? 

4.  How  many  cubic  feet  in  2  cords  1  In  half  a  cord,  how 
many  1     How  many  in  a  quarter  of  a  cordl 

5.  How  many  cubic  yards  in  54  cubic  feet  ?     In  81  ? 

6.  In  120  feet  of  round  timber,  how  many  tons'? 

7.  How  many  tons  of  shipping  in  84  cubic  feet  1     In  168  ? 

8.  How  many  cords  of  wood  in  64  cord  lieeil  In  96  ?  In 
128  ] 

9.  How  many  cubic  feet  in  a  stone  8  feet  long,  3  feet 
wide  and  2  feet  thick  1 

110.  For  what  is  cubic  measure  used  ]  What  are  its  denominations  ? 
>What  is  a  cord  of  wooiH  What  is  a  cord  foot  1  What  is  a  cube? 
What  is  a  cubic  foot  1  What  is  a  cubic  yard  1  How  many  cubic  feet 
in  a  cubic  yard  1  What  are  the  contents  of  a  solid  equal  to  ?  Repeat 
the  table.     What  are  the  units  of  the  scale  % 


106 


REDUCTION    OF 


EXAMPLES. 


1.  In  15r?/.  yd.  18c?/.  ft. 
\^Kn.  in.,  how  many  cubic 
inches  ? 


cu.  yt 

15 

27 

113 

31 


OPERATION. 

I.    cu.ft.  CU.  in, 
18  16 


423  X  1728  =  730960  cw.m, 


2.  In  730960  cubic  inch 
OS,  how  many  cubic  yards, 
&c.  1 

OPERATION. 

1728)730960  cu.  in. 

"27)423" cu.  ft. 

15  cu,  yd. 

cu.  yd.     cu.ft.  cu.  in. 
Ans.     15  18  16 


3.  How  many  small  blocks  1  inch  on  each  edge  can  be 
sawed  out  of  a  cube  7  feet  on  each  edge,  allowing  no  waste 
for  sa\ving  ? 

4.  In  25  cords  of  wood,  how  many  cord  feet  1  How  many 
cubic  feet  1 

5.  How  many  cords  of  wood  in  a  pile  28  feet  long,  4  feet 
wide,  and  6  feet  in  height "? 

6.  In  174 904  cord  feet,  how  many  cords  ? 

7.  In  7645900  cubic  inches,  how  many  tons  of  hewn 
timber  ] 

WINE  OR  LIQUID  MEASURE. 

111.  Wine  measure  is  used  for  measuring  all  liquids  except 
ale,  beer,  and  milk. 

TABLE. 


4  gills,  yi,             make 

1  pint, 

marked         pi. 

2  pints 

1  quart. 

■          -          qt. 

4  quarts 

1  gallon, 

gal. 

31^  gallons 

1   barrel, 

bar.  or  bbl. 

42  gallons 

1  tierce. 

tier. 

63  gallons    • 

1  hogshead 

hhd. 

2  hogsheads 

1  pipe. 

pi. 

2  pipes  or  4  hogsheads 

1  tun,    - 

tun. 

111.  What  is  measurrd  by  wine  or  liquid  measured  What  are  its 
denominations  !  Repeat  the  table.  What  are  the  units  of  the  scale  ? 
VV'hat  is  the  standard  wine  gallon  ? 


DKMOMINATE   NUM23EKS. 


107 


gl.          pt. 

qt. 

gal.      b 

ir.  tier. 

hhd.  pi 

4         =1 

8         =2 

=  1 

32       =8 

=  4 

=  1 

1008   =252 

=  126 

=  311   = 

1 

1344   =336 

=  168 

=  42 

—  1 

2016   =504 

=  252 

=  63 

=  \\ 

=  1 

4032   =1008 

=  504 

=  126 

=  3 

=  2  =  1 

8064  =2016 

=  1008 

=  252 

=  6 

=  4  =  2 

1 

NoTE.-^The  standard  nnit,  or  gallon  of  liquid  measure,  in  the 
United  States,  contains  231  cubic  inches. 

1.  How  many  gills  in  4   pints'?     How  many  pints   in  3 
quarts  ?     In  6  quarts  ?     In  9  ?     In  10  ? 

2.  How   many  quarts  in  2  gallons  ?     In  4  gallons  ?     In  6 
gallons?     How  many  pints  in  2  gallons'?     In  5  ? 

3.  How  many  barrels  in  a  hogshead  1     How  many  in  4 
hogsheads  1     In  6  1 

4.  How  many  quarts  in  3  gallons  1     In  5  gallons?     In  20? 
In  a  barrel  how  many  ?     In  a  hogshead  how  many  ? 


EXAMPLES. 


1 .  In  5  tuns  3  hogs 
17    gallons   of    wine, 
many  gallons  ? 

heads 
how 

2.  In    1466 
many  tuns,  6zc 

gallons,  how 

.  ? 

OPERATION. 

5tuns  3hhd 

.  17 

gal. 

OPERATION. 

63jl466 

4 
23 
63 

4)23 
5 

17  gal. 
"3  hhd. 

76 

139 

Ans.   5tun. 

3hhd.  17  gal 

1466  gallons. 

3.  In  1  2  pipes  1  hogshead  and  1  quart  of  wine,  how  many 
pints  ? 

4.  In  10584  quarts  of  wine,  how  many  tuns  ? 

5.  In  201632  gills,  how  many  tuns? 

6.  What  will  be  the  cost  of  3  hogsheads,  1  barrel,  8  gal- 
lons, and  2  quarts  of  vinegar,  at  4  cents  a  quart  ? 


103 


REDUOTIOW   OF 


ALE  OR  BEER  MEASURE. 

112.  Ale  or  Beer  Measure  is  used  for  measuring  ale,  beer, 
and  milk. 

TABLE. 

make  1 


2  pints,  pt. 

4  quarts 
36  gallons 
54  gallons 
pt. 

2  = 

8  z= 

288         = 
482  = 


qt. 

1 

4 

144 

216 


quart, 
gallon, 
barrel,     - 
hogshead, 
gal.  bar. 


marked  qt. 
gal. 


bar. 
hhd. 
hhd. 


1 
36 
54 


Note. 


=    1 


•1  gallon  contains  282  cubic  inches. 

1.  How  many  pints  in  3  quarts?     How  many  in  5  ? 

2.  How  many  quarts  in  3  gallons?     In  4  gallons?     In  91 

EXAMPLES. 


1.   How  many  quarts  are 
there  in  Vihd.  2bar.  29gal. 
3qt.  ? 

OPERATION. 

2.   In    1271   quarts,  how 
many  hogsheads,  &c.  ? 

OPERATION. 

Ahhd.    2har.  29gal.  3qt. 

H 

4)1271 

2 

4 
4 

36)317       3qt. 
11)8       29gal. 

8bar. 

4          2bar. 

36 

/ 

57 
26 

Am.  4hhd.  2bar.  2gal.  2qt, 

3\7gal. 
4 
1271.7/s. 

3.  In  47 bar. 

4.  In  27  hhd 

5.  In  55832 

6.  In  64972 

IQgal  4qt.,  he 
.  3bar.  26gal.  : 
pints,  how  mar 
quarts,  how  m^ 

)w  many  pints  ? 
^qt.,  how  many  pints? 
ly  hogsheads? 
my  barrels  ? 

112.  For  what  is  ale  or  beer  measure  usedl     What 
uatioiis  ]     Rcj>cat  the  table      What  are  the  scaler  "i 


its  denuiiii- 


DENOMINATE    NUMBERS. 


109 


DRY  MEASURE. 

113.  Dry  Measure   is  used    in   measuring  all  dry  articles, 
such  as  ffrain,  fruit,  salt,  coal,  &:c. 


2  pints,  pL. 

8  quarts    - 

4  pecks 
36  bushels 

pt.' 
2 
16 


TABLE. 

make     1   quart,     marked 
1   peck,    - 


1   bushel, 
1   chaldron, 
pk. 


qt. 
pk. 
bu. 
ch. 


bu. 


ch. 


64 

2304 


=   1. 


=   1 

=   36 

In  5  ]     In  8  ? 

In  32?     In  64? 

In  8?    In  12?     How 
In  40? 


qt 
z    1 

=   8  =1 

z   32  =4 

=   1152        =   144 

1.  How  many  quarts  in  2  pecks  ? 

2.  How  many  pecks  in  24  quarts  1 

3.  How  many  pecks  in  6  bushels  ? 
many  bushels  in  16  pecks  ?     In  32  ? 

4.  How  many  bushels  in  2  chaldrons  ?     In  3  ?     In  4  ? 

Note. — The  standard  bushel  of  the  United  States  is  the  Win- 
chester bushel  of  England.  It  is  a  circular  measure.  18-^  inches  in 
diameter  and  8  inches  deep,  and  contains  21 50|  cubic  inches,  nearly. 

2.  A  gallon,  dry  measure,  contains  268^  cubic  inches. 

EXAMPLES. 

1 .  How  many  quarts  are 
there  in  Qoch.  20bu.  opk. 
Iqt.  ? 


OPERATION. 


^5ch.  20bu.  3pk.  Iqt. 
36 


390 
197 
"2360 

4 

9443 


7o551    quarts. 


&c. 


How  many  chaldrons, 
in  75551  quarts  ? 


OPERATION. 

8)75551 

4)9443 

36)2360 

65 


Iqt. 
3pk. 

2Qbu. 


Ans.  65ch.  20bu.  opk.  Iqt 


113.  What  articles  are  measured  by  dry  measure  1     What  are  its 
denominations  1     Repeat   the  table.     What  are  the  scales  1     What  is 
the  standard  busliel  !     What  are  the  contents  of  a  gallon  \ 
8 


no  KKDUCTION    OF 

3.  In  372  bushels,  how  many  pints  1 

4.  In  6  chaldrons  31   bushels,  how  many  pecks? 

5.  Ill  J 7408  pints,  how  many  bushels? 

6.  In  4220  pints,  how  many  chaldrons  ? 

AVOIRDUPOIS   WEIGHT. 

114.  By  this  weight  all  coarse  articles  are  weighed,  such 
as  hay.  grain,  chandlers'  wares,  and  all  metals  except  gold 
and  silver. 

TABLE. 

16  drams,  dr.    make   1  ounce,     marked    oz, 
16  ounces     -  -        1   pound,  -  lb. 

25  pounds    -  -        1   quarter,        -  qr, 

4  quarters  -        1   hundred  weight,  cwt. 

20  hundred  weight       1  ton, 

dr.  oz.  lb, 

16      =    1 
256      =16  =1 

6400      =   400  =   25 

25600      =    1600  =    100 

512000     =   32000        =   2000        =80       =20        =1 
Notes. — 1.  The  standard  avoirdupois  pound  is  the  weight  of 
27.7015  cubic  inches  of  distilled  water. 

2.  By  the  old  method  of  weighing,  adopted  from  the  English 
system,  112  pounds  were  reckoned  for  a  hundred  weight.  But  now, 
the  laws  of  most  of  the  States,  as  well  as  general  usage,  fix  the 
hundred  weight  at  100  pounds. 

3.  The  units  of  the  scale,  in  passing  from  drams  to  ounces,  are 
16;  from  ounces  to  pounds,  16;  from  pounds  to  quarters,  25; 
from  quarters  to  hundreds,  4;  and  from  hundreds  to  tons,  20. 

1.  In  2oz.,  how  many  drams  ?     In  3  ?     In  4  ?     In  5  ? 

2.  In  4/6.,  how  many  ounces'?       In  3  how  many  ?     In  2  I 

3.  In  Qtqr.,  how  many  hundred  weight  ]     In  5qr.'\ 

4.  In  Zc'wt.,  how  many  quarters  ?     How  many  in  Acwt.  ? 

5.  In  60  hundred  weight,  how  many  tons  ]     In  80  ? 

114.  For  what  is  avoirdupois  weijiht  used]  How  is  /he  table  to  b* 
read  1  How  can  you  detennhie,  from  the  second  table,  the  \alue  ol 
any  unit  in  units  of  the  lower  denominations  ? 


.,    - 

T. 

qr. 

cwt. 

=    1 

=   4    . 
=   80 

=   20 

DKNOMINATK    NUMUIORS. 


Ill 


EXAMPLES. 


1.  How  many  pounds  are 
there  in  15T.  ScwL  3qr. 
X6lb.  ? 

OPERATION. 

15T.  8cwL  3qr.  15/6. 
_20 

SOScwt. 
4' 


1235  qr. 
25 


6180 
2471 

30b90  lb. 


5  lb.  added. 
1  ten  added. 


2.  In  30890  pounds,  hew 
many  tons  ? 

OPERATION. 

25)30890 
4)1235  qr. 


i5lb. 
Sqr. 
15  T.  Scwt, 


20)308  cwt 


15  T.  SctvL  3qr.  15Ib 


3.  In  5T.  Scwt.  3qr.  2Hb.  \3oz.  14^/r.,  how  many  drams? 

4.  In  287^.  Acwt.  Iqr.  21/6.,  how  many  ounces'? 

5.  In  27903G6  drams,  how  many  tons  ? 
G.   In  903 136  ounces,  how  many  tons? 

7.  In  3124446  drams,  how  many  tons? 

8.  In  93  7'.  \3"wt.  3qr.  8lb.,  how  many  ounces? 

9.  In  108910592  drams,  how  many  tons'? 

10.  What  will  be  the  cost  of  117\  l7cwL  3qr.  24/6.  of  hay 
at  half  a  cent  a  pound  ?     How  much  would  that  be  a  ton  1 

11.  What  is  the  cost  of  2T.  \3cwt.  3qr.  21/6.  of  beef  at 
8  cents  a  pound  ?     How  much  would  that  be  a  ton  ? 

TROY  WEIGHT. 

115.  Gold,   silver,  jewels,   and    liquors,    are    weighed  by 

Troy  weight. 

TABLE. 

24  grains,  gr.         make  1   pennyweight,  marked  pwt. 
20   pennyweights       -        1   ounce     -  -  -     oz. 

12  ounces  -         -       1  pound     -         •         -lb, 

gr.  pwt.  oz.  lb. 

24  =    1 

480  =20  =1 

5760  =240  =12  =1 


1V2 


REDUCTION   OF 


Notes. — 1.  The  standard  Troy  pound  is  the  weight  of  22.794377 
cubic  inches  of  distilled  water.  It  is  less  than  the  pound  avoirdupois. 

2.  The  units  of  the  scale,  in  passing  from  grains  to  penny- 
weights, are  24  ;  from  pennyweights  to  ounces,  20 ;  and  from 
ounces  to  po\inds,  12. 

1.  How  many  grains  in  2  pennyweights  ?     In  3  ?     In  41 

2.  How  many  pennyweights  in  48  grains?     In  72  1 

3.  How  many  ounces  in  40  pennyweiglits  ?     In  60  ? 

4.  How  many  ounces  in  4  pounds'?    In  12  ?    In  9  ?    In  7  ? 

5.  How  many  pounds  in  24  ounces  1     In  36  ?     In  96  1 


EXAMPLES. 


1 .  How  many  grains  are 
therein    IQlb.   lloz.  15j)wt. 

OPERATION. 

16/6.  lloz. 

12 


15pwt. 


203 
20 


ounces. 


4075  pennyweights. 
24 


97817  grains. 


2.  In   97817  grains, 
many  pounds  ? 


how 


OPERATION, 

24)07817 
20)4075  pwt. 
12)2203  02. 
■       16  Z6. 


1 6pwt^ 
lloz. 


Ans.  IQib.  lloz.  Idj^wt.  11  gT 


3.  In  25lb.  9oz.  20gr.,  how  many  grains  ? 

4.  In  6490  grains,  how  many  pounds  1 

6.   In  148340  grains,  bow  many  pounds? 

6.  In  11716.  9oz.  I5pwt.  ISgr.,  how  many  grains? 

7.  In  8794  pwt.,  how  many  pounds  1 

8.  In  6Ib.  9oz.  21  gr.  how  many  grains  1 

9.  In  lib.  loz.  lOj^wt.  16«-r.,  how  many  grains  ? 

10.  A  jewel  weighing  2oz.  lApwt.  I8gr.,  is  sold  for  half  a 
dollar  a  grain  :  what  is  its  value  1 


Notes.   1. — What  is  the  standard  avoirdupois  pound  1 

2. — What  is  a  hundred  weight  by  the  English  method!  What  is  a 
hundred  weight  by  the  United  States  method  ] 

3.  Name  the  units  of  the  scale  in  passing  from  one  denomination  to 
another. 

115.  What  articles  are  weighed  by  Troy  weight  1  What  are  its  de- 
nominations ]  Repeat  the  table.  W'hat  is  the  standard  Troy  pound  1 
Wliut  are  the  units  of  the  scale,    in  passiui;  from  one  unit  to  another  1 


DENOMINATE   NUMBERS. 
APOTHECARIES'  WEIGHT. 


118 


116.  This  weight  is  used  by  apothecaries  and  physicians 
in  mixing  their  medicines.  But  medicines  are  generally  sold, 
in  the  quantity,  by  avoirdupois  weight. 

TABLE. 


20  grains,  gi.  make  1  scruple,  marked 

9. 

3  scruples     -     ■ 

1  dram,    -     -     - 

5. 

8  drams         -     • 

1  ounce,    -     -     - 

!. 

12  ounces       -     • 

1  pound,   •     -     • 

fc. 

gr.                   B 

3                  5 

20              =    1 

60              =3 

=    1 

480            =   24 

=   8           =1 

6760          =   288 

=96         =12 

=  1 

Notes. — 1.  The  pound  and  ouikce  are  ihe  same  as  the  pound 
and  ounce  in  Troy  weight. 

2.  The  units  of  the  scale,  in  passing  from  grains  to  scruples, 
are  20 ;  in  passing  from  scruples  to  drams,  3 ;  from  drams  to 
ounces,  8  ;  and  from  ounces  to  pounds,  12. 

1.  How  many  grains  in  2  scruples  1     In  3  ?     In  4  *?    In  6  ? 

2.  How  many  scruples  in  4  drams  1     In  7  drams  1     In  51 

3.  How  many  drams  in  5  ounces  ?  How  many  ounces  in 
32  drams? 


EXAMPLES. 


1.  How   many  grains  in 
9fe   8  5    6  3    2  9    12^r. 

OPERATION. 

9fe  8!   6  3   2  9   12^r. 
12 

116  ounces. 

8 

934  scruples. 
_^ 
*\^04  drams. 

20_ 

600^.2  grains. 


2.  In   56092  grains,  how 
many  pounds  ? 


OPERATION. 

20)56092 

3)2804  9 

12^r. 

8)934  3 

29 

12)116! 

65 

9fe 

8! 

Ans.  9fe  8  5   6  3  2  9  I2gr. 


lU 


REDUCTION    OF 


In271fe95    63    J9,  how  many  scruples  ? 

In  94)fe  115    13,  how  many  drams] 

8011  scruples,  how  many  pounds? 

In  9113  drams,  how  many  pounds'? 

How  many  grains  inl2}fe  9  5    73   29   18^r.  ? 

In  73918  grains,  how  many  pounds  ? 


MEASURE  OF  TIME. 

117.  Time  is  a  part  of  duration.  The  time  in  which  the 
earth  revolves  on  its  axis  is  called  a  day.  The  time -in  which 
it  goes  round  the  sun  is  365  days  and  6  hours,  and  is  called  a 
year.     Time  is  divided  into  parts  according  to  the  following 

TABLE. 

m. 

hr. 

da, 

wk. 

mo, 

y^' 

yr. 
yr 


=   1 

Notes. — 1.  The  years  are  numbered  from  the  beginning  of  the 
Christian  Era.  The  year  is  divided  into  12  calendar  months, 
numbered  from  January  :  the  days  are  numbered  from  the  begin 
ninii  of  the  month :  hours  from  12  at  night  and  12  at  noon. 


60  seconds,  sec.         make 

1  minute,         marked 

60  minutes 

. 

1   hour, 

24  hours     - 

. 

1   day,    - 

7  days 

. 

1   week. 

4  weeks    - 

. 

1  month, 

iomo.  Ida.  and  Ohrs. 
or  365da.  G/ir. 

t 

1  Julian  year. 

12  calendar  months 

• 

1  year, 

sec.                    m. 

hr.             da. 

wk. 

CO                  =   1 

3600              =   60 

= 

1 

86400            =    1440 

— 

24         =    1 

* 

604800          =    10080 

— 

168       =   7 

=   1 

31557600     =   525960 

= 

8766      =   3651 

=   52 

Names. 

No. 

January,     - 

-     1st. 

February,  - 

-     2d. 

March,  -     - 

.     3d. 

April,    .     - 
May,     -     - 
June,     -     - 

.     4lh. 
.     5th. 
.     6th. 

No.  days. 

-  -  31 

-  -  28 

-  -  31 

-  -  30 

-  -  31 

-  -  30 


Names, 

No. 

No.  days 

July,     -     - 

-     7th. 

-     -     31 

August,      - 

-     8th. 

-     -     31 

September, 

-     9Lh. 

-     -     30 

October,      - 

.  10th. 

-     -     31 

November 

-   lUh. 

-     -     30 

December, 

-  12th. 

-     -     31 

DENOMINATE   NUMBERS. 


115 


2.  The  length  of  the  tropical  year  is  365d.  ohr.  48m.  ASscc. 
n*^arly;  but  in  the  examples  we  shall  regard  it  as  365d.  6hr. 

3.  Since  the  length  of  the  year  is  365  days  and  6  hours,  the  o(W 
6  hoais,  by  accumulating  for  4  years,  make  1  day,  so  that  every 
fourth  year  contains  366  days.  This  is  called  Bissextile  or  Leap 
Year  The  leap  years  are  exactly  divisible  by  4  :  1852,  1856,  1860, 
Rre  leap  years. 

4.  The  additional  day,  when  it  occurs,  is  added  to  the  month  of 
February,  so  that  this  month  has  29  days  in  the  leap  year. 

Ihirty  days  hath  September, 
April,  June,  and  November; 
All  the  rest  have  thirty-one, 
Excepting  February,  twenty-eight  alone. 

1.  How  many  seconds  in  4  minutes'^     How  many  in  6? 

2.  How  many  hours  in  3  days  ?     How  many  in  5  ?    In  3  ? 

3.  How  many  days  in  6  w^eeks  ?     In  8,  how  many  ? 

4.  How  many  hours  in  1  week  ]  How  many  weeks  in  4:2da,  1 


1.  How  many  seconds  in 
365da.  6hr.  ? 

OPERATION. 

365c/a.  6hr, 
24 


EXAMPLES, 

2.  How  many  days,  &c. 
in  31557600  seconds  ? 

OPERATION. 

60)31557600 


1466 
730 
8766 
60 


525960  X60=:31557600sec. 


60)525960 
24)8766 
365" 


6Ar. 


Ans.  365da.  ijkr. 


3.  If  the  length  of  the  year  were  365cfa,  237^r.  57m,  39sec 
how  many  seconds  would  there  be  in  12  years'? 

4.  In  126230400  seconds,  how  many  years  of  365  days'? 

5.  In  756952018  seconds,  how  many  years  of  365  days'? 


117.  What  are  the  denominations  of  time?  How  long  is  a  year? 
How  many  days  in  a  common  year1  How  many  days  in  a  Leap  year ! 
How  njany  calendar  months  in  a  year  ?  Name  them,  and  the  number 
of  days  in  each.  How  many  days  has  February  in  the  leap  year!  How 
J>»  \o\i  Tfineiiilifr  whicfi  of  ihu  nunitlis  have  !M)  dayp,  and  which  <l\  "^ 


116  REDUCTION    OF 

6.  Ill  2S5290205  seconds,  how  many  years  of  S65da.  6hr. 
each? 

7.  How  many  hours  in  any  year  from  the  3 1st  day  of  March 
to  the  1st  day  of  January  following,  neither  day  named  being 
counted  ? 

CIRCULAR  MEASURE. 

118.  Circular  measure  is  used  in  estimating  latitude  and 
longitude,  and  also  in  measuring  the  motions  of  the  heavenly 
bodies. 

The  circumference  of  every  circle  is  supposed  to  be  divided 
into  360  equal  parts,  called  degrees.  Each  degree  is  divifled 
into  60  minutes,  and  each  minute  into  60  seconds. 

TABLE. 

60  seconds"         make  1  minute,      marked  '. 

60  minutes  -  1  degree,    -         -  ®, 

30  degrees  -  1  sign,         -         •  8. 

12  signs  or  360°  1  circle,      -         -  o, 

"  '  °  s.  o. 

60  =  1 

3600  =60  =1 

108000  r=  1800  ==30     z=  1 

1296000  =  21600  =360    =12    =1 

1 .  How  many  seconds  in  3  minutes  ?     In  4  ?     In  51 

2.  How  many  minutes  in  6  degrees'?     In  4  ?     In  5  ? 

3.  How  many  degrees  in  4  signs  ?     In  6  ?     In  7  ?     In  8  1 

4.  How  many  degrees  in  240  minutes  ?  In  720  ]  How 
many  signs  in  90°  1     In  150°  ?     In  180°  ? 

EXAMPLES. 

1 .  In  5s.  29°  25',  how  many  minutes  1 

2.  In  2  circles,  how  many  seconds  ? 

3.  In  27894  seconds,  how  many  degrees,  &c. 

4.  In  32295  minutes,  how  many  circles,  (Sec. 

5.  In  3  circles  16°  20',  how  many  seconds? 

6.  In  8s.  16°  25",  how  many  seconds? 

7.  In  8589  seconds,  how  many  degrees,  &c. 

118.  For  wliat  ia  cii'cular  inejwure  used  ?     How  is  every  <Ji«lf  i:-n\> 


DENOMINA'] 


IER6. 


MISCELLANEOI 


12  units,  or  things 

12  dozen     -         -         - 

12  gross,  or  144  dozen 

20  things  - 
100  pounds  - 
196  pounds  - 
200  pounds  - 

18  inches    -         -         - 

22  inches, nearly 

1 4  pounds  of  iron  or  lead 
21  i  stones    -         -         - 


make 


8  pigs 


i^J^dnf^ 


1  great  gioss. 

1  score. 

1  quintal  of  fish, 

1  barrel  of  flour. 

1  barrel  of  pork. 

1  cubit. 

1  sacred  cubit. 

1  stone. 

1  pig- 
1  Ibther. 


BOOKS  AND  PAPER. 


The  terms,  folio^  quarto^  octavo^  duodecimo,  (fee,  indicate 
the  number  ol"  leaves  into  which  a  sheet  of  paper  is  folded. 
A  sheet  folded  in    2  leaves  is  called  a  folio. 
A  sheet  folded  in    4  leaves 
A  sheet  folded  in     8  leaves 
A  sheet  folded  in  12  leaves 
A  sheet  folded  in  16  leaves 
A  sheet  Iblded  in  18  leaves 
A  sheet  folded  in  24  leaves 
A  sheet  folded  in  32  leaves 
24  sheets  of  paper 
20  quires 
2  reams 
6  bundles 

MISCELLANEOUS    EXAMPLES. 

1.  How  many  hours  in  344tf^.  &da.  Mhr.l 

2.  In  6  signs,  how  many  minutes'? 

3.  In  15  tons  of  hewn  timber,  how  many  cubic  inches? 

4.  In  171360  pence,  how  many  pounds'? 

5.  In  1720320  drams,  how  many  tons? 

6.  In  65799  grains  of  laudanum,  how  many  pounds? 

7.  In  9739  grains,  how  many  pounds  Troy? 

8.  In  59lb    \3pwt.  5gr.,  how  many  grains? 

9.  In  £85  85.,  how  many  guineas  ? 
U).  In  346  ii.  F.,  liow  many  Ell?  English'? 


"       a  quarto,  or  4to. 

"       an  octavo,  or  8vo. 

"       a  12mo. 

"       a  16mo. 

"       an  18mo. 

"       a  24mo. 

"       a  32mo. 

ce              1  quire 

1   ream. 

1  bundle. 

1   bale. 

118  KEDITCTTON   OF 

11.  In  'dhhd.  ISgal.  2gt.,  how  many  half-pints  ? 

12.  In  12T.  \5civt.  \qr.  19lb.  I2dr',  how  many  drams? 

13.  In  40144896  square  inches,  how  many  acres  1 

14.  In  5760  grains,  how  many  pomids  ? 

15.  In  6  years  (of  52  weeks  each),  o2wk.  6da.  17/*/-.,  how 
nany  hours? 

16  In  811480'',  how  many  signs  ? 

17  In  2654208  cubic  inches,  how  many  cords'? 

16.  In  18  tons  of  round  timber,  how  many  cubic  inches? 

19.  In  84  chaldrons  of  coal,  how  many  pecks  1 

20.  In  302  ells  Enghsh,  how  many  yards? 

21.  In  24:hhd.  18gal.  2(]L  of  molasses,  how  many  gills? 

22.  In  76-4.  \R.  8P.,  how  many  square  inches  ? 

23.  In  £15  195.  11^.  2>far.,  how  many  farthings? 

24.  In  445577  feet,  how  many  miles  ? 

25.  In  37444325  square  inches,  how  many  acres? 

26.  If  the  entire  surfoce  of  the  earth  is  found  to  contain 
791300159907840000  square  inches,  how  many  square  miles 
are  there  ? 

27.  How  many  times  will  a  wheel  16  feet  and  6  inches  in 
circumference,  turn  round  in  a  distance  of  84  miles  ? 

28.  What  will  28  rods,  129  square  feet  of  land  cost  at  $12 
a  square  foot  ? 

29.  What  will  be  the  cost  of  a  pile  of  wood  36  feet  long, 
6  feet  high  and  4  feet  wide,  at  50  cents  a  cord  foot  ? 

30.  A  man  has  a  journey  to  perform  of  288  miles.  He 
travels  the  distance  in  12  days,  travelling  6  hours  each  day  : 
at  what  rate  does  he  travel  per  hour  ? 

31.  How  many  yards  of  carpeting  1  yard  wide,  will  carpet 
a  room  18  feet  by  20? 

32.  If  the  number  of  inhabitants  in  the  United  States  is 
24  millions,  how  long  will  it  take  a  person  to  count  them, 
counting  at  the  rate  of  100  a  minute  ? 

33.  A  merchant  wishes  to  bottle  a  cask  of  wine  containing 
126  gallons,  in  bottles  containing  1  pint  each  :  how  many 
bottles  are  necessary  ? 

34.  There  is  a  cube,  or  square  piece  of  wood,  4  feet  each 
way  :  how  many  small  cubes  of  1  inch  each  way,  can  be 
sawed  from  it,  allowing  no  waste  in  sawing  ? 

35.  A  merchant  wishes  to  ship  285  bushels  of  flax-seed  in 
casks  containing  7  bushels  2  pecks  each  :  what  number  of 
(!.!ifel<*  MTO  leqiiin"!  ^ 


DKKOMINATE   NUMBEKS.  119 

36.  How  many  times  will  the  wheel  of  a  car,  10  feet  and 
6  inches  in  circumference,  turn  round  in  going  from  Hartlbrd 
to  New  Haven,  a  distance  of  34  miles  ? 

37.  How  many  seconds  old  is  a  man  who  has  lived  32 
years  and  40  days  '? 

38  There  are  15713280  inches  in  the  distance  from  N't  w 
York  to  Boston,  how  many  miles  ? 

39.  What  will  be  the  cost  of  3  loads  of  hay,  each  weighing 
ISctvt.  3qr.  24/6.,  at  7  mills  a  pound? 

AbDITION  OF  DENOMINATE  NUMBERS. 

119.  Addition  of  denominate  numbers  is  the  operation  oi 
finding  a  single  number  equivalent  in  value  to  two  or  more 
given  numbers.     Such  single  number  is  called  the  sum. 

How  many  pounds,  shillings,  and  pence  in  X4  8s.  9d.f 
£27  145.  Ik/.,  and  Xl56  175.  10^/.? 

Analysis. — We  write  the  units  of  the  same  operation. 

Qanie  in  the  same  column.      Add   the  column  £,      g,     cl, 

of  pence  ;  then   30  pence  are  equal  to  2  siiil-  4      8      9 

lings  and  6  pence  :  write  down  the  6,  carrying  oj    14    11 

the  tM'o  to  the  shillings.     Find  the  sum  or'  the  ^Tp    ^-    ^  p. 

shillings,  which  is  41 ;  that  is.  2  pounds  and  1 


shilling  over.  Write  down  Is.  ;  then,  carrying       Xl89      Is.   (JcL 
the   2   to  flie  column  of  pounds,  we  find  the 
Bum  to  be  £189  l5.  6d. 

Note. — In  simple  numbers,  the  number  of  units  of  the  scale, 
at  any  place,  is  always  10.  Hence,  we  carry  1  for  every  10.  In 
denominate  numbers,  the  scale  varies.  The  number  of  units,  in 
pafssing  from  pence  to  shillings,  is  12;  hence,  we  carry  one  for 
every  12.  In  passing  from  shillingstopounds.it  is  20  ;  hence,  we 
carry  one  for  every'  20.  In  passing  from  one  denomination  to 
another,  we  carry  1  for  so  many  units  as  are  contained  in  the  scale 
at  that  place.  Hence,  for  the  addition  of  denominate  numbers,  we 
have  the  following 

Rule. — I.  Set  down  the  riumbers  so  that  units  of  Ji^ 
same  name  shall  stand  in  the  same  column  ; 

II.  Add  as  in  simple  numheis,  and  carry  from  one  de- 
nomination to  another  according  to  the  scale. 

Proof. — The  same  as  in  simple  numbers. 

119.  What  is  addition  of  denominate  numbers'?  How  do  you  set 
down  the  immbers  for  addition  1  How  do  you  adJI  How  do  you 
pi uve  addlliuu  1 


L5S0 

ADDITION  OK 

, 

EXAMPLES. 

,  ( 

1) 

( 

;2.) 

( 

S.) 

£ 

s. 

d. 

£ 

s.     d. 

£ 

..  d 

173 

13 

5 

705 

17  31 

104 

18  9. 

87 

17 

7| 

354 

17  2f 

404 

17  8. 
11  10-- 
14  4 

75 

18 

7^ 

175 

17  3J 

467 

25 

17 

8-i- 

87 

19  7J 
12  7| 

597 

10 

10  lOl 

52 

22 

18  5 

373 

18 

3 

18 

6  5 

TROY  WEIGHT. 

(4.) 

(5.) 

lb. 

OS. 

JTWt. 

gr. 

( 

lb       oz. 

^z/;^.  gr. 

KM 

IOC 

1   10 

19 

20 

171   6 

13  14 

43k 

I       6 

0 

5 

391   11 

9  12 

8C 

1   3 

2 

1 

230   6 

6  13 

^   0 

0 

9 

94   7 

3  18 

)  11 

10 

23 

42  10 

15  20 

1   0 

8 

9 

ARIES'  WEK 

31   0 

0  21 

APOTHEC 

GtHT. 

1 

(6.) 

(7.) 

(8.) 

% 

5 

5 

9  ^^. 

!   3   9 

gr^ 

3  9  gf 

24 

7 

2 

1   1€ 

» 

11  2  1 

17 

3  2  15 

17 

11 

7 

2  1£ 

1 

7  4  2 

14 

0  1  13 

36 

6 

5 

0   7 

r 

4  0  1 

19 

2  2  11 

15 

9 

7 

1  12 

2  5  2 

11 

7  0  17 

9 

3 

4 

1   « 

1     10  1  2  16 
>IRDUP01S  WEIGHT.' 

5  2  14 

AVC 

(9-) 

(10 

.) 

AVt 

.  qr. 

lb. 

02. 

dr. 

T. 

cwt.  qr 

.  lb.     oz. 

14 

2 

0 

14 

9 

15 

12  1 

10  10 

13 

2 

20 

1 

15 

71 

8  2 

6   0 

9 

3 

6 

7 

3 

83 

19  3 

15   5 

10 

0 

18 

12 

11 

36 

7  0 

20  14 

7 

3 

2 

3 

2 

47 

11  2 

2  11 

6 

1 

19 

8 

1 

63 

5  2 

19   7 

4 

3 

0 

15 

5 

12 

13  1 

14   9 

12 

2 

0 

0 

13 

9 

7  0 

5  10 

DEMOMlNATJi    NUilBiaiS.  121 

11.  A  merchant  bought  4  barrels  of  potash  of  the  following 
wei|;hts,  viz.  :  1st,  Zcwt.  2qr.  Olb.  \2oz.  Mr.  ;  2d,  ^civt.  \qr, 
21/6.  4.0Z. ;  3d,  Acwt. ;  4th,  ^cwt.  Oqr.  2lb.  l5oz,  IGdr  '. 
what  was  the  entire  weight  of  the  four  barrels  ? 


LONG  MEASURI 

(12 
X.    mi. fur 
16     2     7 

rd.  yd.  ft, 
39     9     2 

(13.) 
r^Z.    ^cZ.  ft.    in. 
16       9     2     11 

327'    1     2 

20     7     1 

12     11      1       9 

87     0     1 

15     6     1 

18     14     0       7 

1      1     1 

1     2     2 

EASIJR 

19     15     2       1 

CLOTH  M 

E. 

(14.) 

E.  Fl.  qr,  na. 

126     4     4 

(15.) 
yd.  qr. 
4     3 

na. 

2 

^.  11.  qr.  na.  m. 

128     5     1     3 

65     3     1 

5     4 

1 

20     3     1     2 

72     1    ^3 

6     1 

0 

19     1     4     1 

157     2     3 

25     2 

2 

15     3     1     2 

LAND  OR  SQUARE  MEASURE. 

(17.) 
Sq.yd.    Sq.ft. 
97           4 

Sq.  in. 
104 

M. 
2 

(18.) 
A.      R.     P.  Sq.  yd. 
60-3       37     25 

22           3 

27 

6 

375       2       25     21 

105           8 

2 

7 

450       1       31      20 

37           7 

127 

11 

30       0       25     19 

19.  There  are  4  fields,  the  1st  contains  12^.  2R.  38P. ; 
ihe  2d,  4A.  IR.  26P.  ;  the  3d,  85^.  OR.  19P. ;  and  the 
4th,  51  A.  IR.  2 P.  :  how  many  acres  in  the  four  fields'? 


CUBIC  MEASURE. 

(20.) 

(21.) 

(22.) 

Cu,  yd. 

Cu.ft. 

Cu.  in. 

C.     S.ft. 

C. 

Cwdfeet 

65 

25 

1129 

16     127 

87 

9 

37 

26 

132 

17       12 

26 

7 

50 

1 

1064 

18     119 

16 

6 

22 

19 

17 

37     104 

19 

6 

122  AIJDITION   OF 

WINE  OR  LIQUID  MEASURE. 

...       (?^-)  (^^-^ 

kha.   gal.   qU  pt.  tun.  pi.  hhd.  gal.  qt 


127     65     3     2 

14     2 

1     27     3 

12     60     2     3 

15     1 

2     25     2 

450     29     0     1 

4     2 

1     27     1 

2J 

L       0     2     3 

5     0 

1     62     3 

14     39     1     2 

MEAJ 

7     1 

2     21     2 

DRY 

>URE. 

ch. 
27 

(25.) 
hu.  pk.  qt.pt. 
25     3     7      1 

ch. 
141 

(26.) 
hu.  pk.  qt.pt. 
36     3     7     2 

69 

21     2     6     3 

21 

32     2     4     1 

2 

12     7     1 

85 

9     10     3 

5 

9     18     2 

TIME. 

10 

4     4     13 

yr. 

(27.) 
mo.  wk.  da.  hr. 

wk.  da. 

(28.) 
hr.    m.    sec. 

4 

11     3     6     20 

8     8 

14     55     57 

3 

10     2     5     21 

10     7 

23     57     49 

5 

8     1     4     19 

20     6 

14     42     01 

101 

9     3     7     23 

6     5 

23     19     59 

55 

8     4     6     17 
CIRCULAR  MEASUR 

2     2 

20     45     48 

E  OR  M 

OTION. 

8. 

(29.) 

o         /           ti 

s. 

(30.) 

O              /               ft 

5 

17     36     29 

6     29     27     49 

7 

25     41     21 

8     18     29     16 

8 

15     16     09 

7     09     04     58 

Note. — Since  12  signs  make  a  circumference  of  a  circle,  we 
write  down  only  the  excess  over  exact  12's. 


APPLICATIONS    IN    AIDITION. 


1.   Add  46lh.  9oz.    15/;w;^    legr.,   87lb.  lOoz.  6pwt.   14^r., 
lOUM.  lOa-ir.  lOpwt.  lOyr.,  and  C)UO  opwt.  G//r    loncther. 


DENOMINATE    NUMUJCltS.  123 

2.  "What  is  the  weight  of  forty-six  pounds,  eight  ounces, 
thirteen  pennyweights,  fourteen  grains  ;  ninety-seven  pounds, 
three  ounces  ;  and  one  hundred  pounds,  five  ounces,  ten  pen- 
nyweights and  thirteen  grains  ? 

3.  Add  the  following  together:  29 T.  16ctoL  Iqr.  Ulb, 
VZoz.  Mr.,  \Scwt.  Sqr.  lib.,  50 T.  3qr.  4:0Z.,  and  2T.  Iqr. 
Udr. 

4.  What  is  the  weight  of  39 7".  lOcwt.  2qr.  2lb.  15o2.  12dr^^ 
\lcwt.  Qlb.,  12civt.  3qr.,  and  2qr,  8lb.  9dr.  ? 

5.  What  is  the  sura  of  the  following  :  SUA.  2R.  39P. 
20e«^.  ft.  136.vy.  in.,  16^.  IR.  20P.  lOsq.  ft.,  SB.  36P. 
and  4: A.  IE.  16P.  ? 

6.  What  is  the  solid  content  of  64ton  SSft.  800m.,  9ion 
1200m.,  25ft.  700m.,  and  95lon  Sift.  1500m. 

7.  Add  together,  96bu.  Spk.  2qt.  Ipt.,  466m.  Spk.  \qt.  Ipt., 
2pk.  \qt.  Ipt.  and  2Sbu.  Spk.  4qt.  Ipt. 

8.  What  is  the  area  of  the  four  following  pieces  of  land ; 
the  first  containing  20^.  SB.  15P.  250sq.ft.  llQsq.  in.  ;  the 
second,  19 A  IB.  S9P.  ;  the  third,  2B.  lOP.  60sq.ft.  ;  and 
the  fourth,  5A.  6P.  50sq.  in.  1 

9.  A  farmer  raised  from  one  field  S7bu.  \ph.  Sqt.  of  wheat ; 
from  a  second,  416m.  2/>'A-.  5qt.  of  barley  ;  from  a  third,  356/^. 
Ijik.  Sqt.  of  rye  ;  from  a  fourth,  436m.  Spk.  \qt.  of  oats  ;  how 
much  grain  did  he  raise  in  all  ? 

10.  A  grocer  received  an  invoice  of  Ahhd.  of  sugar;  the 
first  weighed  llcwt.  15/6.  ;  the  second,  ]2cwt.  Sqr.  15/6. ;  the 
third,  9ciot.  Iqr.  16/6.  ;  the  fourth,  12cm;/.  Iqr.  :  how  much 
did  the  four  weigh  ? 

11.  A  lady  purchased  32 «/o?5.  3$'r6\  of  sheeting  ;  Slpds.  Iqr. 
of  shirting  ;  li^/ds  2qrs.  of  linen  ;  and  (5yds.  2qrs.  of  cambric: 
what  was  the  whole  number  of  yards  purchased  ? 

12.  Purchased  a  silver  teapot  weighing  2302.  \lpwt.  llgr, ; 
a  sugar  bowl,  weighing  802.  ISpiut.  19 gr.  ;  a  cream  pitcher, 
weighing  5oz.  llgr.  :  what  was  the  weight  of  the  whole? 

.  13.  A  stage  goes  one  day,  87?7i.  6far.  24r{/.  ;  the  next,  75w 
Sfiir.  llrd.)  the  third,  SOm.  Ifar.  \Qrd.  ;  the  fourth,  78 w. 
5fii.r.  :  how  far  does  it  go  in  the  four  days  ? 

14.  Bought  thr^e  pieces  of  land  ;  the  first  contained  17 
uc?'€s  IB.  S5rd.  ;  the  second,  36  acres  2B.  21rd.  ;  and  the 
tliiid.  A<j  acres  OB.  Slrd.  :  how  much  land  did  I  purchafee? 


124 


BUliTK ACTION    OF 


SUBTRACTION  OF  DENOMINATE  NUMBERS. 


OPERATION. 
20    12 

£27     16.S'.   Sd. 
19     17     9 
~1     18   11 


120.  The  difference  between  two  denominate  numbers  is 
Buch  a  number  as  added  to  the  less  will  give  the  greater. 
Subtraction  is  the  operation  of  finding  this  difference. 

I.  What  is  the  difference  between  £27  IQs.  Sd.  and  £19 
17s.  9c/.  1 

Analysis. — We  cannnot  take  %d.  from  Sd.  ; 
we  therefore  add  to  the  upper  number  as  many 
units  as  are  contained  in  the  scale,  and  at  tlie 
same  time  add  1,  mentally,  to  the  next  higher 
denomination  of  the  subtrahend.  We  then  say, 
9  from  20  leaves  1 1 .  Then,  as  we  cannot  sub- 
tract 18  from  16,  we  add  20  and  say.  18  from  36  leaves  18.  Now, 
as  we  have  taken  1  pound  — 20  shillings,  from  the  pounds,  and 
added  it  to  the  shillings,  there  are  but  26  pounds  left.  We  may 
then  say,  19  from  26  leaves  7,  or  20  from  27  leaves  7.  The  lat- 
ter is  the  easiest  in  practice. 

The  first  step  is  called  borrowing^  the  second,  carrying  :  hence, 

Rule. — I.  Set  down  the  less  number  under  the  greater^ 
placing  units  of  the  same  value  in  the  same  column. 

II.  Begin  with  the  lowest  denomination,  and  subtract  as  in 
simple  numbers,  borrowing  and  carrying  for  each  ojjeration 
according  to  the  scale. 

Proof. — The  same  as  in  simple  numbers. 


EXAMPLES. 


A. 

E. 

P. 

From       -     18 

3 

28 

Take       -     15 

2 

30 

Remainder     3 

0 

38 

Proof       -      18 

3 

28 

(2.) 
T.  cwt.  qr.    lb. 
4     12     3     20 
2     18     3 
1     14     0 


■^,\ 


4     12     3     20 


lb.  oz.  pwt.    gr. 

From       .     273  0  0       0 

T'lke       -       98  10  18     21 
Rcinuiiuler 


(4.) 
Ih.      oz.  jjwt.  gr. 

18       9     10       0 
9     10     15     20 


DENOMINATE   NUMBERS. 


121 


T. 

CLvt.  qr.  to.  oz. 

(6) 
cwt.  qr.  lb.     oz.    dr. 

From       .      7 

14     1     3       6 

14     2     12     10     8 

Take        -      2 

6     3     4     11 

6     3     16     15    3 

Remainder 

T. 

(7.) 
hhd.  gal.  qt.  ft. 

(8.) 
yr.  wk.  da.  hr.  '     " 

From     -     151 

3       50     3     2 

95  25    4  20  A5  50 

Take     r       27 

2       54     3     2 

80  30    6  23  46  56 

Remainder 

OPERATION. 

yr. 

mo. 

da. 

1^50 

8 

8 

1848 

7 

5 

TIME  BETWEEN  DATES. 
121.    To  find  the  time  between  any  two  dates. 

1.  What  time  elapsed  between  July  5th,  1848,  and  August 
8th,  1850? 

Note. — In  the  first  date,  the  number  of 
*he  year  is  1848  ;  the  number  of  the  month, 
7,  and  the  numljer  of  the  day,  5  In  the 
second  date,  the  number  of  the  year  is  1850, 

the  number  of  the  month  8,  and  the  number — 

of  the  day,  8.  <J      1      3 

Hence,  to  find  the  time  between  two  dates  : 

Write  the  numbers  of  the  earlier  date  under  those  of  the 
later,  and  subtract  according  to  the  preceding  rule. 

Note, — 1.  In  finaing  the  difference  between  dates,  as  in  casting 
interest,  the  month  is  regarded  as  the  twelfth  part  of  a  year,  and 
as  containing  30  days. 

2.  The  civil  day  begins  and  ends  at  12  o'clock  at  night. 

2.  What  is  the  difierence  of  time  between  Maich  2d, 
1847,  and  July  4th,  1856? 

3.  What  IS  the  difference  of  time  between  April  28th,  1834, 
and  February  3d,  1856? 

4.  What  time  elapsed  between  November  29th,  1836,  and 
January  2d,  1854? 

120.  What  is  the  difference  between  two  denominate  numbers  1 
Give  the  rule  for  subtraction.     How  do  you  prove  subtraction  \ 

121.  Give  the  rule  for  finding  the  difference  between  two  dates.  How 
is  the  month  reckoued  \     At  what  time  does  a  civil  day  be^^iti  1 


126  SUUTKACTION   OF 

5.  What  time  elapsed  between  November  8th,  at  11  o'clock 
A.M.,  1847,  and  December  16th,  at  4  o'clock,  P.M.,  1850  ? 


Analysis. — The  hours  are  numbered  from 
12  at  night,  when  the  civil  day  begins.  The 
numbers  oi  the  years^  months,  days  and  hours, 
are  used. 


OPERATION. 

yr.    mo.     da. 

hr. 

1850    12     16 

16 

1847   11      8 

11 

6.  What  time  elapsed  between  October  9th,  at  11  P.M., 
1840,  and  February  6th,  at  9  P.M.,  1853  ? 

7.  Mr.  Johnson  was  born  September  6th,  1771,  at  9  o'clock 
A.M.,  and  his  first  child  November  5th,  1801,  at  9  o'clock 
P.M. :  what  was  the  difference  of  their  ages  ? 

APrLICATIONS    IN    ADDITION    AND    SUBTRACTION. 

1.  From  38wo.  2wk.  oda.  Ihr.  10m.,  take  lOmo.  3wk. 
2da.  lOhr.  50ni, 

2.  From  1762/r.  8mo.  3wk.  Ada.,  take  91yr.  9mo.  2wk 
6da. 

3.  From  £3,  take  35. 

4.  From  211).,  take  20 gr.  Troy. 

5.  From  8ife,  take  1  lb  1  !    2  3   2  9  . 

6.  From  9T.,  take  IT.  \cwt.  2qr.  20lb.  15oz.  Udr. 

7.  From  3  miles,  take  Sfur.  19rd. 

8.  The  revolution  commenced  April  19th,  1775,  and  a 
general  peace  took  place  January  20,  1783  :  how  long  did 
the  war  contiime  1 

9.  America  was  discovered  by  Columbus,  October  11, 
1492  :  what  was  the  length  of  time  to  July  25,  1855  ? 

10.  I  purchased  167/6.  8oz.  l^pwt.  lOgr.  of  silver,  and 
Bold  98/<^.  lOoz.  \22owt.  19gr.  :  how  much  had  I  left? 

11.  I  bought  19T.  11  cwt.  2qr.  2lb.  \2oz.  \2dr.  of  old 
iron,  and  sold  11  T.  IScwt.  2qr.  \9lb.  Uoz.  lOdr.  :  what  had 
Heft? 

12.  I  purchased  lOlfe  IH  7  3  2  9  19^r.  of  medicine, 
and  Bold  171fe2§  33  IB  5gr.  :  how  much  remained  un- 
sold 1 

13.  From  4:6yd.  \qr.  3na.,  take  A2yd.  Sqr,  \na.  2in. 

14.  Bought  7  cords  of  wood,  and  2  cords  78  feet  havni^ 
been  stolen,  how  much  lemains? 


DENOMINATE   NUMBERS.  l37 

15.  A  owes  B  XlOO  :  what  will  remain  due  after  he  has 
paid  him  £25  35.  6\d.  ? 

16.  A  farmer  raised  136  bushels  of  wheat;  if  he  sells 
4t9bu.  2pk.  Iqt.  \pt.,  how  much  will  he  have  left  1 

17.  From  17 Ahhd.  lOgal.  Iqt.  IjJt.  of  beer,  take  8(Jhhd, 
17 gal.  2qL  Ipi. 

18.  A  farmer  had  57Gbu.  \pk.  2qt.  of  wheat  ;  he  sold 
139/»z^.  2pk.  3qt.  \pt.  :  how  much  remained  unsold? 

19.  A  merchant  bought  17 cwt.  2qr.  \4lb.  of  sugar,  of 
which  he' sold  at  one  time  ocivt.  2qr.  20/6.  ;  at  another  QcwL 
Iqr.  6lh.  :  how  much  remained  unsold  l 

20.  Sold  a  merchant  one  quarter  of  beef  for  £2  75.  9d.  : 
one  cheese  for  95.  7d.  :  20  bushels  of  corn  for  £4'10.v.  lid.  ; 
and  40  bushels  of  wheat  for  £19  125.  8^d.  :  how  much  did 
the  who^e  come  to"? 

21.  Bought  of  a  silversmith  a  teapot,  weighing  316.  Aoz. 
9pwt.  2\gr.  ;  one  dozen  of  silver  spoons,  weighing  2lb.  \o^ 
Ipwt.  ;  2  dishes  weighmg  16lb.  lOoz.  IdpwL  l^gr.  :  how 
much  did  the  whole  weigh  i 

22.  Bought  one  hogshead  of  sugar  weighing  9cwt.  3qr.  2lb. 
14o^r. ;  one  barrel  weighing  3cwt.  \qr.  2lb.,  and  a  second 
barrel  weighing  3cwL  Oqr.  lib.  4.oz. :  how  much  did  the 
whole  weigh  ? 

23.  A  merchant  buys  two  hogsheads  of  sugar,  one  weighing 
6uivt.  oqr.  21/6.,  the  other  9ctct.  2qr.  Qlb.  ;  he  sells  two 
barrels,  one  weighing  3cwt.  Iqr.  12/6.  14oz.,  the  other,  2cwt. 
3qr.  15/6.  6oz.  :  how  much  remains  on  hand  1 

24.  A  man  sets  out  upon  a  journey  and  has  200  miles  to 
travel  ;  the  first  day  he  travels  9  leagues  2  miles  7  furlongs 
30  rods  ;  the  second  day  12  leagues  1  mile  1  furlong  ;  the 
third  day  14  leagues;  the  fourth  day  15  leagues  2  miles 
5  furlongs  35  rods  :  how  far  had  he  then  to  travel  ? 

25.  A  farmer  has  two  meadows,  one  containing  9.4.  3R. 
37 P.,  the  other  contains  10 A    2R.   25 P.;  also  three   pas- 
tures, the  hrst  containing  12^.  IR.  IF.,  the  second  contain 
ing  13^.  3R.,  and  the  third  6^.  IR.  39P.  :  by  how  many 
acres  does  the  pasture  exceed  the  meadow  land  1 

26.  !Supp(?sing  the  Declaration  of  Independence  to  have 
been  published  at  precisely  12  o'clock  on  the  4th  of  July, 
1776,  how  much  time  elapsed  to  the  1st  of  January,  1833 
at  25  minutes  past  3,  P.  M.  ? 


128  MULTIPLICATION   OF 


MULTIPLICATION  OF   DENOMINATE  NUMBERS. 

122.  Multiplication  of  denominate  numbers  is  the  opera- 
tionof  multiplying  a  denominate  number  by  an  abstract  number. 

1.  A  tailor   has   5  pieces  of  cloth  each   containing   ^yd. 
2qr.  3na.  :  how  many  yards  are  there  in  all  ? 

Analysis. — In    all    the    pieces    there    are    5  operation. 

times  as  much  as  there  is  in    1  piece.     If  in      yd.     qr.      na 
1    piece  each  denomination   be  taken   5    times,         6        2        3 
the  result  will  be  5  times  as  great  as  the  multi-  q 

plicand.     Taking    each  denomination  5    times,       — 

we  have  SOyd.  lOqr.  Iowa.  30      10      15 

Rut,  instead  of  writing  the  separate  products,  33  1  3 
we  begin  with  the  lowest  denomination  and 
say,  5  times  Sna.  are  1 5na.  •  divide  by  4,  the  units  of  the  scale,  write 
down  the  remainder  3wa.,  and  reserve  the  quotient  3qr.  for  the 
next  product.  Then  say.  5  times  2(/r.  are  \Oqr.,  to  which  add  the 
3qr.  making  \3qr.  Then  divide  by  4,  write  down  the  remainder 
1,  and  reserve  the  quotient  3  for  the  next  product.  Then  say,  5 
times  6  are  30,  and  3  to  carry  are  33  yards  :  hence, 

IIuLE. — I.  Write  doivn  the  denominate  number  and  set 
the  multiplier  under  the  lowest  denomination. 

1  (,  Multiply  as  in  simple  nuynhers,  and  in  passing  from  one 
d/^iominaiion  to  another,  divide  by  the  unit^  of  the  scale,  set 
d^  wn  the  remainder  and  carry  the  cfKotient  to  the  next  product. 

Proof. — The  same  as  in  simple  numbers. 

EXAMPLES. 

^        (1-)  (2.) 

X       5.       d.  far.  T.  cwt.  qr.  lb.    oz. 

17     15       9     3                          10     0  2     12 

6  7 


106     14     10     2  3     10     0     19     4 

(3.)  (4.) 

m.fur.  rd.  yd.  ft,  s.     °      '        " 

9     3     20     3     2  9     9     27     35 

6  3 


i32.  What  is  multu)lication  of  denominate  numbers  1     Give  tlie  rule 
How  do  you  prove  JiiuIli[»licati(Mi  1 


DENOMINATE   NUMBERS.  129 

(5.)  (6.) 

yr.  mo  da.    hr.  T.  cwt.  qr.    lb.  oz.  dr. 

6     5     15     18  6     12     3     20  12     9 
5  8 


7.  A  farmer  has  11  bags  of  corn,  each  containing  2bu.  \pk, 
Zqt.  :  how  much  corn  in  all  the  bags  ? 

8.  How  m^uch  sugar  in  12  barrels,  each  containing  ^cwt. 
3^r.  2lbA 

9.  In  '7  loads  of  wood,  each  containing  1  cord  and  2  cord 
feet,  how  many  cords  ? 

10.  A  bond  was  given  21st  of  May,  1825,  and  was  taken 
up  the  12th  of  March,  1831  ;  what  will  be  the  product,  if 
the  time  which  elapsed  irom  the  date  of  the  bond  till  the  day 
it  was  taken  up  be  multiplied  by  3  l 

11.  What  i?  the  weight  of  1  dozen  silver  spoons,  each 
weighing  3o2..  %pwt.  ? 

12.  What  is  the  weight  of  7  tierces  of  rice,  each  weighmg 
5civt.  2qr.  \6lb.  ? 

13.  Bought  4  packages  of  medicine,  each  containing  3ife 
4  §    6  3    19    1 6gr.  :  what  is  the  weight  of  all  1 

14.  How  far  will  a  man  travel  in  5  days  at  the  rate  of 
24wz.  Afur.  4:rd.  per  day  ? 

15.  How  much  land  is  there  in  9  fields,  each  field  contain- 
ing 12^1.  IR.  25P.? 

16.  How  many  yards  in  9  pieces,  each  29i/d.  2q7\  3/m.  1 

17.  11"  a  vessel  sails  6L.  2mi.  Qfur.  3ijrd.  in  one  day, 
how  far  will  it  sail  in  8  days  1 

18.  How  much  water  will  be  contained  in  96  hogsheads, 
each  containing  62[/aL  \qt.  ipt.  \gi.  1 

Note. — When  the  multiplier  is  a  composite  number,  and  the 
factors  do  not  exceed  12,  multiply  by  the  factors  in  succeiision. 
In  the  last  example  96=12x8. 

19.  If  one  spoon  weigh  3o2.  6pwt.  I5gr.,  what  is  the 
weight  of  1 20  spoons  ? 

20.  If  a  man  travel  24mi.  Ifur.  ^rd.  in  one  day,  how  far 
will  he  go  in  one  month  of  30  days  ? 

21.  If  the  earth  revolve  0°  15'  of  space  per  minute  of  time, 
how  far  does  it  revolve  per  hour  ? 

22.  Bought  dO/Lhd.  of  sugar,  each  weighing  \2owL  2qr. 
Wlb.  :  what  was  the  weight  ol  the  whole? 

9 


130  DIVISION    OF 

23.  What  is  the  cost  of  18  sheep,  at  5.s-.  9^d.  apiece  ? 

24.  How  much  molasses  is  contained  in  26hhd.  each  hogs- 
head having  6\gal.  \qt.  Ipt.  ? 

25.  How  many  yards  of  cloth  in  36  pieces,  each  piece  con- 
taining 25i/(L  3qr.  1 

26.  A  farmer  has  18  lots,  and  each  lot  contains  41.4.  2R, 
HP.  :  how  many  acres  does  he  own  ? 

27.  There  are  three  men  whose  mutual  ages  are  14  times 
20y?'.  6nio.  owk.  ^da.  :  what  is  the  sum  of  their  ages  ? 

28.  Bought  9Qhhd.  of  sugar,  each  weighing  I2cwt.  2qr, 
14/6.  ;  what  is  the  weight  of  the  whole  ? 

29.  If  a  vessel  sail  49mi.  6fur.  8rd.  in  one  day,  how  far 
will  she  sail  in  one  month  of  30  days  ? 

30.  Suppose  each  of  50  farmers  to  raise  125  bu.  Spk.  6qt  of 
grain  :  how  much  do  they  all  raise  1 

31.  If  a  steam  ship,  in  crossing  the  Atlantic,  goes  2\l77ii. 
4fur.  32rd.  a  day,  how  far  will  she  go  in  15  days  ? 

32.  If  1  horse  consume  2  to7is  Iqr.  20/i.  of  hay  in  a  winter, 
how  much  will  36  horses  consume  1 

33.  How  much  cloth  will  clothe  a  company  of  48  men,  if 
it  takes  5yd.  3qr.  2na.  to  clothe  one  man  % 

Note. — Eacli  denomination  may  be  multiplied  by  the  multiplier, 
sepai-atelyj  and  the  results  reduced  and  multiplied. 

DIVISION  OF  DENOMINATE  NUMBERS. 

123.  Division  of  denominate  numbers  is  the  operation  of 
dividing  a  denominate  number  into  as  many  equal  parts  as 
there  are  units  in  the  divisor. 

1.  Divide  £25  15s-.  4:d.  by  8. 

Analysis. — We  first  say  8  into  25,  3  times  operation. 

and  £1  or  205.  over.     Then,  after  adding  tlie     8)X25    15«.   4c/. 
155.  we  say,  8  into  35,  4  times  and  3s.   over.  T-q      r'    ^ 

Then,  reducing  the  3s.  to  pence  and  adding  in 
the  4<i.,  Ae  say,  8  into  40,  5  times. 

123.  What  is  division  of  denominate  numbers'!     Give   the  rule  for 

division.     How  do  you  prove  division  \  How  do  you  divide  when  the 

divisor  is  a  composite  number  1     What  will  be  the  unit  of  each  quo- 
tient figure  \ 


DLNOMIKATE   NUMliKKS.  .  131 

OPERATION. 

2.  Divide  36^1^.  3;?A:.  Iqt.  by  7.  35  ^         ^   ^ 

Analysis. — In   this   example    we  \ 

find  that  7  is  contained  in  36  bushels  a 

5  times  and  1  bushel  over.  Reducing  — 

this    to   pecks,    and    adding  3  pecks,  l)l'pk.\\j)k. 

gives    7  pecks,  which  contains   7,   1  7 

time  and  no  remainder.  Multiplying  '     q    ^ 

0    by  8  quarts  and   adding,    gives   7  g 

quarts  to  be  divided  by  7.  . 

Ans.  5bu.  \pk.  Iqt. 

Hence,  for  the  division  of  denominate  numbers  we  have  the 
following  • 

Rule. — I.  Begin  laith  the  highest  denomination  and 
divide  as  in  simple  numbers  : 

II.  Reduce  the  remainder,  if  any,  to  the  next  lower  de- 
nomitiation,  and  add  in  the  units  of  that  denominxition  for 
a  neio  dividend. 

III.  Proceed  in  the  same  manner  through  all  the  denorrvi- 
nations  to  tlie  last. 

Proof.— Same  as  in  simple  numbers. 

Notes. — 1.  If  the  divi.sor  is  a  composite  number,  we  may  divide 
by  the  factors  in  succession,  as  in  simple  numbers. 

2.  Eacn  quotient  figure  has  the  same  unit  as  the  dividend  from 
from  which  it  was  derived. 

3.  If  the  divisor  is  greater  than  12  and  not  a  composite  number 
the  operation  is  the  same  as  long  division. 


EXAMPLES. 

T. 

7)1 

cict.  qr. 
19     2 

lb. 
12 

(2.) 
A.     R. 
9)113    3 

P. 
25 

(Quotient. 

0     2 

16 

12    2 

25 

L. 

8)47 

(3.) 

7111.  fur. 

1       7 

,   rd. 

8 

(4.) 
bu.    ph. 
11)25      3 

qt. 

1 

Uuotieiit. 

132  I)IVII*lON   OF 

Divide  tlie  followins: : 


5.  17cwt.0qr,2lk'^oz.hy  7, 

6.  A9i/d.  Sqr.  3?ia.  by  9. 

7.  131A.  IR.  by  12. 


8.  £1138  125.  Ad.  bv  6'^. 

9.  70  2:  llctvL  7Z^.  by  79. 
10.   27bu.  Spk.  Iqt.  by  84. 


11.  Bought  65  yards  of  cloth  for  which  I  paid  £72  '[As. 
A\d.  :   what  did  it  cost  per  yardl 

12.  If  15  loads  of  hay  contain  35T.  5cwt.,  what  is  the 
weight  of  each  load  ? 

13.  If  a  man,  lifting  8  times  as  much  as  a  boy,  can  raise 
201/6.  12o2.,  how  much  can  the  boy  liff? 

14.  If  a  vessel  sail  25°  42'  40"  in  10  days,  how  far  will 
she  sail  in  one  day? 

15.  Divide  9hhd.  2Sgal.  2qf.  by  12. 

16.  What  is  the  quotient  o{  ^5bu.  Ipk.  3qt.  divided  by  12? 

17.  In  4  equal  packages  of  medicine  there  are  13)fe  7  ^ 
2  3     1  ^    4^r.  ;  how  much  is  there  in  each  package  ? 

18.  In  25hhd.  of  molasses,  the  leakage  has  reduced  the 
whole  amount  to  \63Agal.  Iqt.  \pt.  :  if  the  same  quantity 
has  leaked  out  of  each  hogshead,  how  much  will  each  hogs- 
head still  contain  ? 

19.  In  9  fields  there  are  113A  3R.  25F.  of  land:  if  the 
fields  contain  an  equal  amount,  how  much  is  there  in  each 
field? 

20.  If  in  30  days  a  man  travels  lA^mi.  6fur.,  travelling 
the  same  distance  each  day,  what  is  the  length  of  each  day's 
journey  ? 

21.  Suppose  a  man  had  98/6.  2oz.  19pwt.  6gr.  of  silver; 
how  much  must  he  give  to  each  of  7  men  if  he  divides  it 
equally  among  them  ? 

22.  When  \15gal.  2qt.  of  beer  are  drank  in  52  weeks, 
how  much  is  consumed  in  one  week  ? 

23.  A  rich  man  divided  1686w.  \2jk.  6qt.  of  corn  among 
35  poor  men  :  how  much  did  each  receive  ? 

24.  In  sixty-three  barrels  of  sugar  there  are  IT.  I6cwt. 
3qr.  1216.  :  how  much  is  there  in  each  barrel  ? 

25.  A   farmer  has  a  granary   containing    232    bushels  3 
pecks  7  quarts  of  wheat,  and  he  wishes  to  put  it  in  105  bags 
how  much  must  each  bag  contain  ? 

26.  If  90  hogsheads  ol' sugar  weigh  567^.  lAcwt.  3qr.  15//a, 
what  is  the  weight  of  1  hogshead  ? 


«EJNUMUNATE(/^iyiijE]t»/'  E  H  SI  T  TA 

27.   One  hundred  and  seventy h* 
IScwt.  2qr.   15/6.  6oz.  of  bread  :  fi^ 
consume  ^ 

25.  If  the  earth  revolves  on  its  axis  15°  in  1  hour,  how  far 
does  it  revolve  in  1  minute  ? 

29.  If  59  casks  contain  44McZ.  53^al.  2qt.  Ipt.  of  wine, 
what  are  the  contents  of  one  cask  1 

30.  Suppose  a  man  has  2^6mi.  &fur.  2Qrd.  to  travel  in  12 
days  :  how  far  must  he  travel  each  day  ? 

31.  If 'l  pay  £12  145.  5</.  3far,  for  35  bushels  of  wheat, 
what  is  the  price  per  bushel  ? 

32.  A  printer  uses  one  sheet  of  paper  for  every  16  pages  of 
an  octavo  book  :  how  much  paper  wall  be  necessary  to  print 
500  copies  of  a  book  containing  336  pages,  allowing  2  quires 
of  waste  paper  in  each  ream  ?* 

33.  A  man  lends  his  neighbor  £135  65.  Qd.,  and  takes  in 
part  payment  4  cows  at  £5  Ss.  apiece,  also  a  horse  worth 
£50  :  how  much  remained  due  1 

34.  Out  of  a  pipe  of  wine,  a  merchant  draws  12  bottles, 
each  containing  1  pint  3  gills  ;  he  then  fills  six  5-gallon  demi- 
johns ;  then  he  draws  off  3  dozen  bottles,  each  containing 
1  quart  2  gills  :  how  much  remained  in  the  cask  ? 

35.  A  farmer  has  ^T.  Scwt.  2qr.  14^1Ij.  of  hay  to  be  re- 
moved in  6  equal  loads :  how  much  must  he  carried  at  each 
load  ? 

36.  A  person  at  his  death  left  landed  estate  to  the  amount 
of  £2000,  and  personal  property  to  the  amount  of  £2803  175. 
4d.  He  directed  that  his  widow  should  receive  one-eighth  of 
the  whole,  and  that  the  residue  should  be  equally  divided 
among  his  four  children  :  what  was  the  wddow  and  each 
child's  portion  ? 

37.  If  a  steamboat  go  224  miles  in  a  day,  how  long  will 
it  take  to  go  to  China,  the  distance  being  about  12000  milfes  ? 

38.  How  long  would  it  take  a  balloon  to  go  from  the  earth 
to  the  moon,  allowing  the  distance  to  be  about  240000  miles, 
the  balloon  ascending  34  miles  per  hour '? 


*  In  packing  and  selling  paper,  the  two  outside  quires  of  every  ream 
are  regarded  as  waste,  and  each  of  the  remaining  quires  contains  24 
perfect  sheets  :  hence,  in  this  example,  the  waste  paper  is  considered 
ass  belonging  only  to  the  entire  reams 


13ti  LONGITUDE    AND    TIME. 


LONGITUDE  AND  TIME. 


124.  The  circumference  of  the  earth,  like  that  of  othei 
circles,  is  divided  into  360°,  which  are  called  degrees  of  longir 
*'>jide, 

125.  The  sun  apparently  goes  round  the  earth  once  in  24 
hours.     This  time  is  called  a  day. 

Hence,  in  24  hours,  the  sun  apparently  passes  over  360°  of 
longitude  ;  and  in  1  hour  over  360°-^-24=:  15°. 

126.  Since  the  sun,  in  passing  over  15°  of  longitude,  re- 
quires 1  hour  or  60'  of  time,  1°  v^'ill  require  60'-^15=z:4 
minutes  of  time  ;  and  V  of  longitude  will  be  equal  to  one 
sixtieth  of  4'  which  is  4"  :  hence, 

15°  of  longitude  require  1  hour. 
1°  of  longitude  requires  4  minutes. 
1 '  ol'  longitude  requires  4  seconds. 
Hence,  we  see  that, 

1.  If  the  degrees  (f  longitude  he  multiplied  by  4,  the  irro- 
duct  will  be  the  corresponding  time  in  minutes. 

2.  If  ilie  minutes  m  longitude  be  ?nultiplied  by  4,  the  j^ro- 
duel  will  be  the  corresponding  time  in  seconds. 

127.  When  the  sun  is  on  the  meridian  of  any  place,  it  ia 
12  o'clock,  or  noon,  at  that  place. 

Now,  as  the  sun  apparently  goes  from  east  to  west,  at  the 
instant  of  noon,  it  will  be  past  noon  for  all  places  at  the  east, 
and  before  noon  for  all  places  at  the  west. 

If  then,  we  find  the  difierence  of  time  between  two  places 
and  know  the  exact  time  at  one  of  them,  the  corresponding 
time  at  the  other  will  be  found  by  adding  their  difierence,  if 
that  other  be  east,  or  by  subtracting  it  lowest. 

124.  How  is  the  circumference  of  the  earth  supposed  to  be  divided  ] 

125.  How  does  the  sun  appear  to  move  !  What  is  a  day  1  How  far 
does  the  sun  appear  to  move  in  1  hour  \ 

126.  How  do  you  reduce  degrees  of  longitude  to  timel  How  do  you 
reduce  minutes  of  longitude  to  time  "^ 

127.  What  is  the  hour  when  the  sun  is  on  the  meridian  ■;  When  the 
sun  is  on  the  meridian  of  any  place,  how  will  the  time  be  for  all  places 
east !  How  for  all  places  west !  If  you  have  the  difl'erence  of  time  how 
do  you  find  the  time'' 


Longitude  and  timk.  135 

1.  The  longitude  of  New  York  is  74°  1'  west,  and  that  of 
Philadelphia  75°  10'  west:  what  is  the  difference  of  longi- 
tude and  what  their  difference  of  timel 

2.  At  12  M.  at  Philadelphia,  what  is  the  time  at  New 
York  ? 

3.  At  1 2  M.  at  New  York,  what  is  the  time  at  Philadelphia  ? 

4.  The  longitude  of  Cincinnati,  Ohio,  is  84°  24'  Avest : 
what  is  the  diiierence  of  time  between  New  York  and  Cin- 
cinnati ? 

t).  What  is  the  time  at  Cincinnati,  when  it  is  12  o'clock  at 
New  York  1 

6.  The  longitude  of  New  Orleans  is  89°  2'  west  :  what 
time  is  it  at  New  Orleans,  when  it  is  12  M.  at  New  York  ? 

7.  The  meridian  from  which  the  longitudes  are  reckoned 
passes  through  the  Greenwich  Observatory,  London  :  hence, 
the  longitude  of  that  place  is  0  :  what  is  the  difference  of 
time  between  Greenwich  and  New  York  1 

8.  What  is  the  time  at  Greenwich,  when  it  is  12  M.  at 
New  York  ? 

9.  The  longitude  of  St.  Louis  is  90°  15'  west  :  what  is  the 
time  at  St.  Louis,  when  it  i^  3k.  25'  P.  M.  at  New  York  ? 

10.  The  longitude  of  Boston  is  71°  4'  west,  and  that  of 
New  Orleans  89°  2'  west  :  what  is  the  time  at  New  Orleans 
when  it  is  7  o'clock  12'  A.  M.  at  Boston  ? 

11.  The  longitude  of  Chicago,  Illinois,  is  87°  30' west  • 
what  is  the  time  at  Chicago,  when  it  is  12  M.  at  New  York  ? 

PROPERTIES  OF  NUMBERS. 

COMPOSITE    AND    PRIME    NUMBERS. 

128.  An  Integer,  or  whole  number,  is  a  unit  ©r  a  collection 
of  units. 

129.  One  number  is  said  to  be  divisible  by  another,  when 
the  quotient  arising  from  the  division  is  a  whc^e  number.  The 
division  is  then  said  to  be  exact. 

Note — Since  every  number  is  divisible  by  itself  and  1,  the 
term  divisible  will  be  applied  to  such  numbers  orzZi/,  as  have  other 
divisors. 

128.  What  is  an  uiteger  1 


lob  PKOPEKTIES    OF   NUMBEKS. 

130.  Every  divisible  number  is  called  a  comjoosite  nunibej* 
(Art.  54),  and  any  divisor  is  called  di  factor:  thus,  6  is  a  com- 
posite number,  and  the  factors  are  2  and  3. 

i31.  Every  number  which  is  not  divisible  is  called  a  prime 
number:  thus,  1,  2,  3,  5,  7,  11,  &c.  are  prime  rmmbers. 

132.  Every  prime  number  is  divisible  by  itself  and  1  ; 
but  since  these  divisors  are  common  to  all  numbers,  they  are 
not  called /ac^OTA-. 

133.  Every  factor  of  a  number  is  either  prime  or  compo- 
site :  and  since  any  composite  factor  may  be  again  divided,  it 
follows  that. 

Any  number  is  equal  to  the  product  of  all  its  prim^e  factors. 

For  example,  12  =  6x2  ;  but  6  is  a  composite  number,  of 
which  the  factors  are  2  and  3  ;  hence, 

12=2x3x2;  also,  20  =  10x2=5x2x2. 

Hence,  to  find  the  prime  factors  of  any  number, 

Divide  the  number  by  any  pri^e  number  that  will  exactly 
divide  it :  then  divide  the  quotient  by  any  prime  number  that 
will  exactly  divide  it^  and  so  on^  till  a  quotient  is  found  ivhich 
is  a  prime  number  ;  the  several  divisors  and  the  last  quotient 
will  be  the  prime  factors  of  the  given  number. 

Note. — It  is  most  convenient,  in  practice,  to  use  the  least  prime 
number,  which  is  a  divisor. 

1.  What  are  the  prime  factors  of  42  ] 

Analysis.  —  Two  being  the  least  divisor 
that  is  a  prime  number,  we  divide  by  it,  giv- 
ing the  quotient  21,  which  we  again  divide  3)21 
by  3,  giving  7  :  hence,  2.  3  and  7  are  the  Y 
prime  factors. 

2X3X7  =  42. 

129.  When  is  one  number  divisible  by  another  1  By  what  is  every 
number  divisible  1     Is  1  called  a  divisor  1 

130.  What  is  a  composite  number  1     What  is  a  factor  1 
r.U.   What  is  a  prime  number  \ 

132.  By  what  divisors  is  every  prime  number  divided  1 

133.  To  what  product  is  every  number  equal  1  Give  the  rule  fox 
finding  the  prime  factors  of  a  numler.  What  number  is  it  nwst  conve- 
nient to  use  as  a  divisor  1 


OPERATION. 

2)42 


PiilME    FACTOKS.  137 

What  are  the  prime  fabtors  of  the  following  numbers  t 


1.  Of  the  number  9? 

2.  Of  the  number  15  1 

3.  Of  the  number  24? 

4.  Of  the  number  16  ] 
6.  Of  the  number  18? 


6.  Of  the  number  32? 

7.  Of  the  number  48? 

8.  Of  the  number  56? 

9.  Of  the  number  63  ? 
10.  Of  the  number  76? 


Note. — The  prime  factors,  when  the  number  is  small,  may 
generally  be  seen  by  inspection.  The  teacher  can  easily  multiply 
the  examples. 

134.  When  there  are  several  numbers  whose  prime  factors 
are  to  be  found, 

I^ind  the  prime  factors  of  each  and  then  select  those  factors 
vahich  are  common  to  all  the  numbers. 

11.  What  are  the  prime  factors  common  to  6,  9  and  24  ? 

12.  What  are  the  prime  factors  common  to  21,  63  and  84  ? 

13.  What  are  the  prime  factors  common  to  21,  63  and  106? 

14.  What  are  the  common  ffictors  of  28,  42  and  70? 

15.  What  are  the  prime  factors  of  84,  126  and  210  ? 

16.  What  are  the  prime  factors  of  210,  315  and  525? 

135.    DIVISIBILITY    OF    NUMBERS. 

1.  2  is  the  only  even  number  which  is  prime. 

2.  2  divides  every  even  number  and  no  odd  number. 

3.  3  divides  any  number  when  the  sum  of  its  figures  is  di 
visible  by  3. 

4.  4  divides  any  number  when  the  number  expressed  by 
the  two  right  hand  figures  is  divisible  by  4. 

5.  5  divides  every  number  which  ends  in  0  or  5. 

6.  6  divides  any  even  number  which  is  divisible  by  3. 

7.  10  divides  any  number  ending  in  0. 

GREATEST  COMMON  DIVISOR. 

136.  The  greatest  common  divisor  of  two  or  more  num- 
bers, is  the  greatest  number  which  will  divide  each  of  them, 
separately,  without  a  remainder.  Thus,  6  is  the  greatest 
common  divisor  of  12  and  18. 

134.  IIovv  do  you  fnul  the  prime  factors  of  two  or  moie  umubersi 


138  COMMON    DIVISOK. 

Note. — Since  1  divides  every  number,  it  is  not  reckoned  among 
the  common  divisors. 

137.  If  two  numbers  have  no  common  divisor,  they  are 
called  prime  with  respect  to  each  other. 

138.  Since  a  factor  of  a  number  always  divides  it,  it  fol- 
lows that  the  greatest  common  divisor  of  two  or  more  rmm- 
bers,  is  simply  the  greatest  factor  common  to  these  numbers. 

Hence,  to  find  the  greatest  common  divisor  of  two  or  more 
numbers, 

I.  Resolve  each  number  into  its  p^'ime  factors. 

II.  The  jyroduct  of  the  factors  common  to  each  result  will 
he  the  greatest  comrrwn  divisor. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  24  and  30  ? 

Analysis.  —  There   are    four   prime  opkration. 

factors  in  24,  and  3  in   30  :  the  factors  24  =  2x2x2x3 

2  and  3  are   common:    hence,  6  is  the  30  =  2x3x5 

greatest  common  divisor.  2x3  =  6  com.  divisor 

2.  What  is  the  greatest  common  divisor  of  9  and  18  1 

3.  What  is  the  greatest  common  divisor  of  6,  12  and  301 

4.  What  is  the  greatest  common  divisor  of  15,  25  and  30  ? 

5.  What  is  the  greatest  common  divisor  of  12,  18  and  72  ? 

6.  What  is  the  greatest  common  divisor  of  25,  35  and  70  ? 

7.  What  is  the  greatest  common  divisor  of  28,  42  and  70  1 

8.  What  is  the  greatest  common  divisor  of  84,  126  and 
210  1 

139.  When  the  numbers  are  large,  another  method  of  find- 
ing their  greatest  common  divisor  is  used,  which  depends  on 
the  following  principles  : 

135.  What  even  number  is  prime  1  What  numbeis  will  2  divide  1 
What  numbers  will  3  divide  1  What  numbers  will  4  divide  ?  51  6  ^ 
101 

136.  What  is  the  greatest  common  divisor  of  two  or  more  numbers  ] 

137.  When  are  two  numbers  said  to  be  prime  with  respect  to  each 
other  ] 

138.  What  is  the  greatest  factor  of  two  numbers'!  How  do  you  find 
the  greatest  common  divisor  of  two  or  more  numbers  ? 


PliOFERTIES    OF   NUMBERS.  Iti9 

^.  Any  nuviber  which  tvill  divide  tico  numbers  separately,  will 
divide  their  sum;  else,  tve  shaidd  have  a      04  4-  ^7  —  -51 
ivhole  number  equal  to  a  proper  fraction.  "    "~ 

2.  Any  number  which  ivill  divide  two  numbers  separately, 
ivill   divide   their  difference;   and  any 
number  which  will  divide  their  difftfr-      51  —  27  =z  24 
ence  and  one  of  the  numbers,  ivill  divide 
the  other  ;  else,  tue  should  have  a  whole  number  equal  to  a 
'proper  fraction. 

1.   What  is  the  greatest  common  divisor  of  27  and  51  ? 

Divide  51  by  27  ;  the  quotient  is  1  and  the  remainder  24;  then 
divide  the  preceding  divisor  27  by  the^^^re- 
mainder  24:  the  quotien  is  1  and  there-      t)n\K\i^ 
mainder    3;    then    divide    the    preceding  '^IS 

divisor   24   by  the  remainder  3  ;  the  quo-  ^' 

tient  is  8  and  the  remainder  0.  24)27(1 

iVow,   since   3  divides  the  difference  3,  24 

and  also  24,  it  will  divide  27,  by  principle  ~3T^478 

2d  ;  and  since  3  divides  the  remainder  24,  ^  ,  ^ 

and  27.  it  will  also  divide  51  ;  hence,  it  is  "* 

a  common  divisor  of  27  and  51  ;  and  since  it  is  the  greatest  com- 
mon factor,  it  it  their  greatest  common  divisor.  Since  the  above 
reasoning  is  as  applicable  lo  any  other  two  numbers  as  to  27  and 
5],  wc  liave  the  following  rule  : 

Divide  the  greater  number  by  the  less,  and  then  divide  the 
jyreceding  divisor  by  the  remainder,  and  so  on,  till  nothing  re- 
mains :  the  last  divisor  will  be  the  greatest  common  divisor, 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  216  and  408  ? 

2.  Find  the  greatest  common  divisor  of  408  and  740. 

3.  Find  the  greatest  common  divisor  of  315  and  810. 

4.  Find  the  greatest  common  divisor  of  4410  and  5670. 

5.  Find  the  greatest  common  divisor  of  3471  and  1869. 

6.  Find  the  greatest  common  divisor  of  1584  and  2772  ? 

Note. — If  it  be  required  to  find  the  greatest  common  di\isor  of 
more  than  two  numbers,  first  find   the  greatest  common  divisor  of 

139.  When  the  numbers  are  large,  on  what  principles  does  the  oper- 
ation of  finding  the  greatest  common  divisor  depend  !  What  is  the 
rule  for  thiding  it ! 


140  COMMON    DIVIDEND. 

two  of  tliem,  then  of  that  common  divisor  and  one  of  the  remaiu 
ing  numbers,  and  so  on  for  all  the  numbers :  the  last  commou 
divisor  will  be  the  greatest  common  divisor  of  ail  the  numbers. 

7.  What  is  the  greatest  common  divisor  of  492,  744,  and 
1044  ? 

8.  What  is  the  greatest  common  divisor  of  944,  1488,  and 
2058  1 

9.  What  is  the  greatest  common  divisor  of  216,  408,  and 
740? 

10  W^hat  is  the  greatest  common  divisor  of  945,  1560,  and 
22083  ? 

LEAST  COMMON  DIVIDEND. 

140.  The  least  common  dividend  of  two  or  more  numbers 
is  the  least  number  which  they  will  separately  divide  without 
a  remainder.* 

Notes. — 1.  If  a  dividend  is  exactly  divisible  by  a  divisor,  it  can 
be  resolved  into  two  factors,  one  of  which  is  the  divisor  and  the 
other  the  quotient. 

2.  If  the  divisor  be  resolved  into  its  prime  factors,  the  cor- 
responding factor  of  the  dividend  may  be  resolved  into  the  same 
factors ;  hence,  the  dividend  will  contain  every  prime  factor  of  the 
divisor. 

3.  The  question  of  finding  the  least  common  dividend  of  several 
numbers,  is  therefore  reduced  to  finding  a  number  which  shall  con- 
tain all  their  prime  factors  and  none  others. 

1.  Let  it  be  required  to  fmd  the  least  common  dividend  of 
6,  8  and  12. 

Anai.ysis.— We  see,  from  inspec-                      operation. 
tion,  that  the  prime  factors  of  6,  are          2^3     2X2X2     2X2X3 
2  and  3  ;— of  8  ;  2,  2  and  2  ;— and 
©f  12;  2,  2  and  3.  b  .  .  .  .  » x<i 

Every  number  that  is  a  prime  factor  must  appear  in  the  least  com- 
mon dividend,  and  none  others;  hence,  it  will  contain  all  the  prime 

140  Wbat  is  the  least  common  dividond  of  two  or  more  numbers  1 
State  the  principlps  involved  in  finding  it.  Give  the  rule  for  finding  it. 
What  is  the  dividend  when  the  numbern  have  no  common  prime  fac- 
tors 1 

*  The  number  which  we  call  the  least  common  dividend  is  generally 
willed  the  least  common  multiple.     We  prefer  the  former  fur  beginners 


COMMON    DIVIDEND.  141 

factors  of  any  one  of  the  numbers,  as  8,  and  such  other  prime  fac- 
tors of  the  others,  6  and  12,  as  are  not  found  among  the  prime  fac- 
tors of  8  ;  that  is,  the  factor  3  :  hence, 

2  X  2  X  2  X  3  ::=  24,  the  least  common  dividend. 

To  find  the  least  common  dividend  of  several  numbers, 

1.  Place  the  numbers  on  the  same  line,  and  divide  by  any 
jmme  number  that  will  exactly  divide  two  or  more  of  them, 
and  set  down  in  a  line  below  the  quotients  and  the  undivided 
numbers. 

II.  Then  divide  as  before  until  there  is  no  prime  number 
greater  than  1  that  will  exactly  divide  any  two  of  the  numbers. 

III.  Then  multiply  together  the  divisors  and  the  numbers  of 
the  lower  line,  and  their  product  will  be  the  least  common 
dividend. 

Note — 1.  The  object  of  dividing  by  any  prime  number  that  will 
divide  two  or  more  of  the  numbers,   is  to  find  common  factors. 

2.  If  the  numbers  have  no  common  prime  factor,  their  product 
will  be  their  least  common  dividend. 


EXAMPLES, 


OPERATION. 


2)3  . 

.  .  4  .  . 

.  .  .  8 

2)3  .  . 

,  .  2.  . 

.  .  .  4 

3  .  . 
3)3  .  . 

.  1  .  . 

,.  .  8. 

.  ..  2 
9 

1.  Find  the  least  common  divi- 
dend of  3,  4  and  8. 

Ans.   2x2x3x1x2  =  24. 


2.  Find  the  least  common  divi- 
dend of  3,  8  and  9. 

Ans.  3x1x8x3  =  72.  1  ....  8 3 

3.  Find  the  least  common  dividend  of  6,  7,  8  and  10. 

4.  Find  the  least  common  dividend  of  21  and  49. 

5.  Find  the  least  common  dividend  of  2,  7,  5,  6,  and  8. 

6.  Find  the  least  common  dividend  of  4,  14,  28  and  98. 

7.  Find  the  least  common  dividend  of  13  and  6. 

8.  Find  the  least  common  dividend  of  12,  4  and  7. 

9.  Find  the  least  common  dividend  of  G,  9,  4,  14  and  16. 

10.  Find  the  least  common  dividend  of  13,  12  and  4. 

11.  Find  the  least  common  dividend  of  11,  17,  19,21,  and  9 


112 


CANCELLATION. 


CANCELLATION. 


141.  Cancellation  is  a  method  of  shortening  Arithmeti- 
cal operations  hy  omitting  or  cancelling  common  factors. 

1.   Divide  24  by  12.     First,  24^3x8;  and  12  =  3x4. 

Analysis. — Tweuty-four  divided  by  12  is  operation. 

equal  to  3X8  divided  by  3X4;  by  cancelling         24       ^  x  8 
or  stiiking  out  the  3's,  we  have  8  divided  by        To  —  .v      .  =^ ^' 
4,  wJiich  is  equal  to  2.  i^;       ^  X4 

142.  The  operations  in  cancellation  depend  on  two  princi- 
ples : 

1.  The  cancelling  of  a  factor^  in  any  number^  is  equivalent 
to  dividing  the  nmnher  by  that  factor. 

2.  If  the  dividend  and  divisor  be  both  divided  by  the  same 
number,  the  quotient  will  not  be  changed. 

PRINCIPLES    AND    EXAMPLES. 

1.  Divide  63  by  21. 

Analysis.— Resolve  the  dividend  and  divi-  operation. 

Bor  into  factors,  and  then  cancel  those  which         ^^ _1l  X9 
are  common.  2 1  ~  ^  X  3 

2.  In  7  times  56,  how  many  times  8  ? 

Analysis. — Resolve  56  into  the  operation. 
two  factors  7  and  8,  and  then  cancel  56  X7  ^X7x7 
the  8.  -g        = ^T"" 

3.  In  9  times  84,  how  many  times  12? 

4.  In  14  times  63,  how  many  times  7 

5.  In  24  times  9,  how  many  times  8  ? 

6.  In  36  times  15,  how  many  times  45?  • 
Analysis. — We  see  that  9  is  a  factor  of  36 

and   45.     Divide  by  this  factor,  and  write  the  operation. 

quotient  4  over  36,  and  the  quotient   5  below  4        3 

45.     Again,  5  is  a  factor  of  15  and  5.     Divide        $0  Xl'0_-i2 

15  by   5,   and  write   the  quotient  3   over    15. 

Dividing  5  by  5,  reduces  the  divisor  to  1,  which 

need  not  be  set  down :  hence,  the  true  quotient 

4X3=12. 


==3. 


4^ 


141.  What  is  cancellation  ? 

142-  On  what  du  the  operations  of  cancellation  depend  ? 


CANCELLATION. 


143 


143.   Therefore,  to  perform  the  operations  of  cancellation  : 

1.  Resolve  the  dividend  and  divisor  into  such  factors  as 
shall  give  all  the  factors  common  to  both. 

II.  Cancel  the  cormnon  factors  and  then  divide  the  jjroduci 
of  the  remaining  factors  of  the  dividend  by  the  product  of  the 
remaining  factors  of  the  divisor. 

Notes. — 1.  Since  every  factor  is  cancelled  by  division^  the  quo- 
tient 1  always  takes  tlie  place  of  the  cancelled  factor,  but  is  omit- 
ted when  it  is  a  multiplier  of  other  factors. 

2.  If  one  of  the  numbers  contams  a  laclor  equal  to  the  product  of 
two  or  more  factors  of  the  other,  all  the  factors  may  be  cancelled. 

3.  If  the  product  of  two  or  more  factors  of  the  dividend  is  equal 
to  the  product  of  two  or  more  factors  of  the  divisor,  such  factors 
may  be  cancelled. 

4.  It  is  generally  more  convenient  to  set  the  dividend  on  the 
right  of  a  vertical  line  and  the  divisor  on  the  left. 

EXAMPLES. 

1.  What  number  is  equal  to  36  multiplied  by  13  and  the 
product  divided  by  4  times  9  1 

Analysis. — We  may  place  the  numbers  whose         operation. 
product  forms  the  dividend  on  the  right  of  a  verti- 
cal line,   and  those  which  form  the  divisor  on   the 
left.     We  see  that  4X9  =  36  ;  we  then  cancel  4.  9, 
and  36.  ^''^-  ^^' 

2.  What  is  the  result  of  20x4x12,  divided  by 
10x16x3? 

Analysis. — First,  cancel  the  factor  10,  in  10 
and  20,  and  write  the  quotients  1  and  2  above 
the  numbers.  We  then  see  that  16X3—48.  and 
that  4X  12  =  48  ;  cancel  16  and  3  in  the  divisor. 


13 


OPERATION. 


1 


and  4  and 
tient  is  2. 


.  2  in  the  dividend  ;  hence,  the  quo- 


u 

n 

a:0 

4 

$ 

n 

Am   2. 


3.  Divide  the  product  of  126  X  16  X  3,  by  7  X  12. 


ANALYsis.-^We  see  that  7  is  a  factor 
of  126 — giving  a  quotient  of  18.  We 
cancel  7,  and  place  18  at  the  right  of 
126.  We  then  cancel  6,  in  12  and  18, 
and  write  the  quotients  2  and  3  at  the 
right.  We  then  cancel  the  factor  2,  in 
2  and  16,  and  set  down  the  quotients  1 
and  8.     The  product  of  1X1   is  the  di- 


OPERATION 


1 


% 


n 


m 


8 


Ans.  3x8x3=72. 

vii;or,  and  the  iii-<uluct  of  3X8X3  =  72.  the  dividend. 


144 


CANCELLATION. 


4.   What  is  the  quotient  of  3  X  8  X  9  x7  X  15,  divided  b\i 
63  X  24  X  3  X  5  ?  ^ 


OPERATION. 

03 


Analysis.— The  63  is   cancelled  by  7x9;    24         M 
by  3x8  ;  3  and  5,  by  15;  hence,  the  quotient  is  1.  $ 


3[$ 


5.  Divide  the  product  of  6x7x9x11,  by  2x3x7x3 
X21. 

6.  Divide  the  product  of  4  x  14  x  16  X  24,  by  7x8x32 
X12. 

7.  Divide  the  product  of  5  X  11  X  9  X  7  X  15x  6,  by  30  X  3 
X  21x3x5. 

8.  Divide  the  produ^it  of  6  X  9  X  8  X  1 1  X  12  x  5,  by  27  X  2 
X32x3. 

9.  Divide  the  product  of  1  X  6  x  9  x  14  x  15  x  7  X  8,  by  36 
X  126x56x20. 

10.  Divide  the  product  of  18  x  36  x  72  x  144,  by  6  X  6  x  8 
X9xl2x8. 

11.  Divide  the  product  of  4  X  6  x  3  x  5,  by  5  x  9  x  12  x  16 

12.  Multiply  288  by  16,  and  divide  the  product  by  8x9 
X2x2. 

13.  In  a  certain  operation  the  numbers  24,  28,  32,  49,  81, 
are  to  be  multiplied  together  and  the  product  divided  by 
8x4x7x9x6  :  what  is  the  result  1 

14.  Multiply  240  by  18  and  divide  the  product  by  6 
times  90. 

15.  Divide  16  x  20  x  8  x  3,  by  30  x  8  x  6. 

16.  How  many  pounds  of  butter  worth  15  cents  a  pound, 
may  be  bought  lor  25  pounds  of  tea  at  48  cents' a  pound  1 

'l7.  How  much  calico  at  25  cents  a  yard  must  be  given 
for  100  yards  of  Irish  sheeting  at  87  cents  a  yard "? 

18.  How  many  yards  of  cloth  at  46  cents  a  yard  must  ba 
given  for  23  bushels  of  rye  at  92  cents  a  bushel  i 

i43.  Give  the  nile  for  the  operation  of  cancellation. 


CANCELLATION.  145 

19.  How  many  bushels  of  oats  at  42  cents  a  bushel  must 
be  given  for  3  boxes  of  raisins  each  containing  26  pounds,  at 
14  cents  a  pound  ? 

20.  A  man  l^iys  2  pieces  of  cotton  cloth,  each  containing 
33  yards  at  11  cents  a  yard,  and  pays  for  it  in  butter  at  18 
cents  a  pound  :  how  many  pounds  of  butter  did  he  give  ? 

21.  If  sugar  can  be  bought  for  7  cents  a  pound,  how  many 
bushels  of  oats  at  42  cents  a  bushel  must  1  give  for  56  pounds  ? 

22.  If  wool  is  worth  36  cents  a  pound,  how  many  pounds 
m\ist  be  given  for  27  yards  of  broadcloth  worth  4  dollars  a 
yard  ? 

23.  If  cotton  cloth  is  worth  9  cents  a  yard,  how  much 
must  be  given  for  3  tons  of  hay  worth  1 5  dollars  a  ton  ? 

24.  How  much  molasses  at  42  cents  a  gallon  must  be  given 
for  216  pounds  of  sugar  at  7  cents  a  pound  ? 

25.  Bought  48  yards  of  cloth  at  125  cents  a  yard:  how 
many  bushels  of  potatoes  are  required  to  pay  for  it  at  150 
cents  a  bushel? 

26.  Mr.  Butcher  sold  342  pounds  of  beef  at  6  cents  a 
pound,  and  received  his  pay  in  molasses  at  36  cents  a  gallon  : 
how  many  gallons  did  he  receive  ? 

27.  Mr.  Farmer  sold  1263  pounds  of  wool  at  5  cents  a 
pound,  and  took  his  pay  in  cloth  at  421  cents  a  yard  :  how 
many  yards  did  he  take  1 

28.  How  many  firkins  of  butter,  each  containing  56  pounds, 
at  18  cents  a  pound,  must  be  given  for  3  barrels  of  sugar, 
each  containing  200  pounds,  at  9  cents  a  pound  ? 

29.  How  many  boxes  of  tea,  each  containing  24  pounds, 
worth  5  shillings  a  pound,  must  be  given  for  4  bins  of  wheat, 
each  containing  145  busheh,  at  12  shillings  a  bushel  ? 

30.  A  worked  for  B  8  days,  at  6  shillings  a  day,  for  which 
he  received  12  bushels  of  corn:  how  much  was  the  corn 
worth  a  bushel  1 

31.  Bought  15  barrels  of  apples,  each  containing  2  bushels, 
at  the  rate  of  3  shillings  a  bushel  :  how  many  cheeses,  each 
weighing  30  pounds,  at  1  shilhng  a  pound,  will  pay  for  the 
apples  ? 


10 


146  OOMMCiN    FliACTIONS. 


COMMON    FRACTIONS. 

144.  The   unit    1    denotes    an    entire   thing,    as    I    apple, 

1  chair,  1  pound  of  tea.  ' 

If  the  unit  1  be  divided  into  two  equal  parts,  each  part  is 
called  one-half. 

If  the  unit  1  be  divided  into  three  equal  parts,  each  part  is 
called  one-third. 

If  the  unit  1  be  divided  into  four  equal  parts,  each  part  is 
called  one-fnurlli. 

If  the  unit  1  be  divided  into  twelve  equal  parts,  each  part 
is  called  one-twelflh  ;  and  if  it  be  divided  into  any  number  oi 
equal  parts,  we  have  a  like  expression  for  each  part. 

The  parts  are  thus  written  : 

is  read,  one-seventh. 

-  -       one-eighth. 

-  -       one-tenth. 

-  -       one- fifteenth. 

-  -       one-fiftieth. 

an  entire  third ;  the  i,  an 
entire  fourth  ;  and  the  same  for  each  of  the  other  equal  parts 
hence,  each  equal  part  is  an  entire  thing,  and  is  called  ^fraC' 
tional  unit. 

The  unit,  or  whole  thing  which  is  divided,  is  called  the 
unit  of  the  fraction. 

Note. — In  every  fraction  let  the  pupil  distinguish  carefully 
between  the  unit  of  the  fraction  and  the  fractional  unit.  The  rirst 
is  the  Vfhole  thing  from  which  the  fraction  is  derived  ;  the  second, 
(ne  of  the  equal  parts  into  which  that  thing  is  divided, 

145.  Each  fractional  unit  may,  like  the  unit  1,  become  the 
base  of  a  collection  :  thus,  suppose  it  were  required  to  express 

2  of  each  of  the  fractional  units  :  we  should  then  write 

144.  What  is  a  unit?  What  is  each  part  called  when  the  unit  1  ij» 
divided  into  two  equal  parts  1  When  it  is  divided  into  3 1  Into  4  !  Into 
51   Into  12? 

How  may  the  one-half  be  regarded"!  The  one-third  1  The  on  e- fourth  ? 
What  is  each  part  called  ! 

What  is  the  unit  of  a  fraction  1  What  is  a  fractional  unit  1  How  ao 
you  distinyuish  between  the  one  and  the  (»ther1 


i  is  read. 

one-half. 

1 

i  ■  - 

one-third. 

1 

i  ■  ■ 

one- fourth. 

A- 

*  -  ■ 

one-fifth. 

tV 

i  ■  - 

one- sixth. 

sV 

The  i,  is  an 

entire  half ; 

the  J, 

COMMON     FKACTIONS.  147 


1 
1 

which  is  read 
((        (<       (( 

(<        ((       <( 

2  halves  =-Jx  2 
2  thirds  =JX2 
2  fourths=J-x2 

f 

((        ((        (i 

2  fifths   =ix2 

&-C.,       &;c., 

&:c.,       &c. 

If  it  were  required  to  express  3  of  each  of  the  fractional 
units,  we  should  write 

I       which  is  read  3  halves  =i  x  3 

I  "       "       "  3  thirds   - Jx3 

f  "       "      "  3  fourths=ix3 

f  "       "      "  3  fifths    =ix3 

&c.,       &c.,       &c.,  &c.  ;  hence, 

A  FRACTION  is  a  collection  of  one  or  more  of  the  equal  parts 
of  a  unit. 

Fractions  are  expressed  hy  two  numbers,  the  one  written 
above  the  other,  with  a  line  between  them.  The  lower  num- 
ber is  called  the  denominator,  and  the  upper  number  the 
numerator. 

The  denominator  denotes  the  number  of  equal  parts  into 
which  the  unit  is  divided  ;  and  hence,  determines  the  value 
of  the  fractional  unit.  Thus,  if  the  denominator  is  2,  the 
fractional  unit  is  one-ha/f;  if  it  is  3,  the  fractional  unit  is  (me- 
third  ;  if  it  is  4,  the  fractional  unit  is  one  fourth,  dz;c  ,  &c. 

The  numerator  denotes  the  number  of  fractional  units  taken. 
Thus,  f  denotes  that  the  fractional  unit  is  ^,  and  that  3  such 
units  are  taken  ;  and  similarly  for  other  fractions. 

In  the  fraction  |,  the  base  of  the  collection  of  fractional 
units  is  1,  but  this  is  not  the  primary  base.  For,  \  is  one- 
fifth  of  the  unit  1  ;  hence,  the  primary  base  of  every  fraction 
is  the  unit  1. 

145.  May  a  fractional  unit  become  the  base  of  a  collection  1  What  is 
a  fraction  !  How  are  fractions  expressed  1  What  is  the  lower  numbei 
called  !  What  is  the  upper  number  called  1  What  does  the  denomina- 
tor denote  ?  What  does  the  numerator  denote  1  In  the  fraction 
5)  fifths,  what  is  the  fractional  base  1  What  is  the  primary  base  "^  What 
is  the  primary  base  <if  every  fraction  ! 


148  (X)MMON    FRACTION S. 

146.  If  we  suppose  a  second  unit  of  the  same  kind  to  be 
divided  into  equal  parts,  such  parts  may  be  expressed  in  the 
same  collection  with  the  parts  of  the  first :  thus, 

I  is  read  3  halves. 

I  "     "  7  fourths. 

1/  "     "  16  fifths. 

^^  ♦'     "  18  sixths. 

y  "     "  26  sevenths. 

147.  A  whole  number  may  be  expressed  fractionally  by 
writing  1  below  it  for  a  denominator.     Thus, 

3  may  be  written  ^  and  is  read,  3  ones. 

5  -       -         -  ^  -     -     -       5  ones. 

6  -       -         -  1^  -     -     -       6  ones. 
8     -       -         -  "I  -     -     -       8  ones. 

But  3  ones  are  equal  to  3,  5  ones  to  5,  6  ones  to  6,  and 
8  ones  to  8  ;  hence,  the  value  of  a  number  is  not  changed  by 
placing  1  under  it  for  a  denominator. 

148.  [f  the  numerator  of  a  fraction  be  divided  by  its  de- 
nominator, the  integral  part  of  the  quotient  will  express  the 
number  of  entire  units  used  in  forming  the  fraction  ;  and  the 
remainder  will  show  how  many  fractional  units  are  over. 
Thus,  y  are  equal  to  3  and  2  thirds,  and  is  written  y  =  3|-  : 
hence, 

A  fraction  has  the  same  form  as  an  U7iexecuted  division. 

From  what  has  been  said,  we  conclude  that, 

1st.    A  fraction  is  one  or  more  of  the  equal  parts  of  a  unit 

2d.    The  denominator  shows  into  how  many  equal  parts 

the  unit  is  divided,  and  hence  indicates  the  value  of  the 

fractional  unit  : 

146.  If  a  second  'unit  be  divided  into  equal  parts,  may  the  parts  be 
expressed  with  those  of  the  first  1  How  many  units  have  been  divided 
to  obtain  6  thirds  \     To  obtain  9  halves  ^      12  fourths  1 

147.  How  may  a  whole  number  be  expressed  fractionally''  Does 
this  change  the  value  of  the  number  \ 

148.  If  the  numerator  be  divided  by  the  denominator,  what  does  the 
quotient  show  ]  What  does  the  remainder  t-.how  ]  What  fonn  hae  a 
frikClion  !     What  aic  tho  yeven  prhici|>lf.'ij  w!\iclj  follow  ■? 


COMMON    FRAC'no»^S.  149 

3d.  The  numerator  sliows  Jioiv  many  fractional  units  are 
taken  : 

4th.  The  value  of  every  fraction  is  equal  to  the  quotient 
arising  from  dividing  the  numerator  by  the  denominator  : 

5th.  When  the  nutnerator  is  less  than  the  deno7ni?tator, 
the  value  of  the  fraction  is  less  tJian  1. 

6th.  When  the  numerator  is  equal  to  the  denominator, 
the  value  of  the  fraction  is  equal  to  1. 

7th.  When  the  numerator  is  greater  than  the  denominor 
tor,  the  value  of  the  fraction  is  greater  than  1. 

EXAMPLES    IN    WRITING    AND    READING    FRACTIONS. 


1     Read  the  following  fractions  ; 

5        5       1_6         7         3      _9_        6  5_ 
T2'     9'       7   '    T&'     85     50'    Tl  7 


in  each  example?     How  many  fractional  units  are  taken  in  each? 

2.  Write  12  of  the  17  equal  parts  of  1. 

3.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one-twentieth,  express  6  fractional  units.  Express  12,  18, 
16,  30,  fractional  units. 

4.  If  the  fractional  unit  is  one  36th,  express  32  fractional 
units  ;  35,  38,  54,  6,  8. 

5.  If  the  fractional  unit  is  one-fortieth,  express  9  fractional 
units;  16,  25.  69,  75. 

DEFINITIONS. 

149.  A  Proper  Fraction  is  one  whose  numerator  is  less 
than  the  denominator. 

The  following  are  proper  fractions  : 

111135.        9_85. 
2'     35     4>     4'     7'     8'    TO'     9'     6* 

150.  An  Improper  Fraction  is  one  whose  numerator  is 
equal  to,  or  exceeds  the  denominator. 

Note. — Such  a  fraction  is  called  improper  because  its  value 
equals  or  exceeds  1. 

149.  What  is  a  proper  fraction  1     Give  examples. 

150    What  is  an  improjier  fraction  1     Wl>y  iiniToperl     Give  exam - 


160  riio POSITIONS  uv 

The  following  are  improper  fractions  : 

3       5.      6       8.      9       1_2       1_4       X9 

2'     3'     5'     7'     8'      6   '       7   »       7   ' 

151.  A  Simple  Fraction  is  one  whose  numerator  and  de- 
nominator are  both  whole  numbers. 

Note. — A  simple  fraction  may  be  either  proper  or  improper. 

The  following  are  simple  fractions  : 

i.3.58.££6      1 
4'     2'     6'     7'     2'     3>    3'     5* 

152.  A  Compound  Fraction  is  a  fraction  of  a  fraction,  or 
several  fractions  connected  by  the  word  of. 

The  following  are  compound  fractions  : 

i  of  i,    i  of  i  of  i      i  of  3,     4  of  J  of  4. 

153.  A  Mixed  Number  is  made  up  of  a  whole  number  and 
a  fraction. 

The  following  are  mixed  numbers  : 

3i,         %  6f,         5f,  6f,  3i. 

154.  A  Complex  Fraction  is  one  whose  numerator  or  do- 
norainator  is  fractional ;  or,  in  which  both  are  fractional. 

The  following  are  complex  fractions  : 

4  2  I  45j 

5'  19^'  f  69f 

155.  The  numerator  and  denominator  of  a  fraction,  taken 
together,  are  called  the  ter7ns  of  the  fraction  :  hence,  every 
fraction  has  two  terms. 

fundamental  propositions. 

156.  By  multiplying  the  unit  1,  we  form  all  the  whole 
numbers, 

161,  What  is  a  simple  fraction  1  Give  examples.  May  it  be  proper 
or  improper  ^ 

152.  What  is  a  compound  fraction  1     Give  examples. 

153.  What  is  a  mixed  number  1     Give  examples. 

154.  What  is  a  complex  fraction  1     Give  examples. 

155.  Hov^'  many  terms  has  every  fraction  1     What  are  they  T 

15G.  How  may  all  the  whole  numbers  be  formed  1  How  may  the 
fractional  units  be  formed  1  How  many  times  is  one-half  less  than  1 1 
How  many  thucs  is  auy  fraitiunal  unif  less  than  1  1 


COMMON    FRACTIONS.  151 

2,     3,     4,     5,     6,     7,     8,     9,      10,     &c.  ; 

and  by  dividing  the  unit  1  by  these  numbers  we  orm  ail  the 
fractional  units, 

1      JL      1       I      1       i      JL      1      JL      &e 

2'        3'         4>        5'        6'         7'         8'        9'         10'        *^*^" 

Now,  since  in  1  unit  there  are  2  halves,  3  thirds,  4 
fourths,  5  filths,  6  sixths,  &c.,  it  follows  that  the  fractional 
unit  becomes  less  as  the  denominators  are  increased  :  hence, 

The  fractional  unit  is  such  a  part  of  I,  as  1  is  of  the 
dcncnninator  of  the  fraction. 

Thus,  1  is  such  a  part  of  1,  as  1  is  of  2  ;  J  is  such  a  part  of 
1,  as  1  is  of  3 ;  i  is  such  a  part  of  1  as  1  is  of  4,  ^c.  6lc. 

157.  Let  it  be  required  to  multiply  ^  by  3. 

Analysis. — In  -|  there    are    5    fractional  operation. 

units,  each  of  which  is  ^,  and  these  are  to  |^  X  3=:-^^==-^ 
be    taken   3   times.     But  5   things    taken   3 

times,  gives  15  tilings  of  the  same  kind  ;  that  is,  15  sixths  ;  hence, 
the  product  is  3  limes  as  great  as  the  multiplicand :  therefore,  we 
have 

V  Proposition  I. — If  the  numerator  of  a  fraction  be  mitltiplied 
by  any  number^  the  fraction  will  be  increased  as  many  times  as 
there  are  units  in  the  multiplier. 


EXAMPLES. 


1.  Multiply  I  by  8. 

2.  Multiply  I  by  5. 

3.  Multiply  4  by  9. 


4.  Multiply  ^  by  14. 

5.  Multiply  I  by  20. 

6.  Multiply  Yf  by  25. 


158.  Let  it  be  required  to  multiply  J  by  3. 

Analysis. — In  ^  there  are  4  fractional  operation. 

units,  each  of  which  is  ^.     If  we  divide     -i  X  3  r:z  -4_  —  4 
the  denominator  by  3.  we  change  the  frac- 
tional unit  to  ^,  which  is  3  times  as  great  as  ^.  since  the  first  is 
contained  in  1,  2  times,  and  the  second  6  times.     If  we  take  ibis 
fractional   unit  4  times,  the  result  ^.  is  3  times   as  great  as  |  : 
therefore,  we  have 

Proposition  II. — If  the  denominator  of  a  fraction  be  divi- 
ded  by  any  multiplier,  the  value  of  the  fraction  will  be  in- 
creased as  many  times  as  there  are  u?tits  in  that  multiplier. 

167.  What  is  proved  hi  Proposition  I  \ 


152  PHOi'OSITlO^'S    IN 


EXAMPLES. 


4.  Multiply  If  by  2,  4,  G, 

5.  Multiply  fi  by  2,  6,  7. 

6.  Multiply  i§J  by  5,  10. 


1.  Multiply  I  by  2,  by  4. 

2.  Multiply  if  by  2,  4,  8. 

.3.  Multiply  j%  by  2,  4,  6. 

159.   Let  it  be  required  to  divide  ^j  by  3. 

Analysis. — In  ^j.  there  are  9  fractional  operation. 

niiits^  each  o!"  which  is  ^,  and  these  are  o  _i_3—  »^ir;^. 
to  be  divided  by  3.     But  9  things,  divided 

by  3,  sives  3  things  of  the  same  kind  for  a  quotient ;  hence,  the 
quotient  is  3  elevenths,  a  number  one-third  as  great  as  ^j ;  hence, 
we  have 

Proposition  III. — If  the  numerator  of  a  fraction  he  divi- 
ded by  any  mimber,  ike  value  of  t}i£  fraction  will  he  dimin- 
isited  as  many  times  as  there  are  units  in  the  divisor. 


JXAMPLKS. 


1.  Divide  fl  by  2,  by  7. 

2.  Divide  |i|  by  56. 


3.  Divide  f^f  by  25,  by  8. 

4.  Divide  fl-g  by  8,  16,10. 


160.   Let  it  be  required  to  divide  ^  by  3. 

xVnalysis. — In  y\,  there  are  9  fractional  operation, 

units,  each  of  which  is  ^.  Now,  if  we  ^-^3=j^yx5=^' 
multiply  the  denominator  by  3  it  becomes 

33,  and  the  fractional  unit  becomes  ^,  which  is  only  \  of  ^,  be- 
cause 33  is  3  times  as  great  as  11.  if  we  take  this  fractional 
unit  9  times,  the  result,  ^,  is  exactly  ^  of  -^^ :  hence,  we 
nave 

Proposition  IV. — If  the  denominator  of  a  fraction  be 
taultiplied  by  any  divisor,  the  value  of  the  fraction  will  be 
diminished  as  many  times  as  there  are  units  in  that  divisor. 


1.  Divide  i  by  2. 

2.  Divide  l  by  7. 

3.  Divide  y\  by  4. 


EXAMPLES. 

4.  Divide  ^  by  8. 

5.  Divide^  by  17. 

6.  Divide  ^f^  by  45. 


158.  What  is  proved  in  proposition  II.  1 

159.  What  is  proved  in  proposition  III.1 
ItJO.  What  is  ppDvod  in  propo.'^ition  IV. ! 


(X>MM:on    FRAOl'lONS.  153 

161.  Let  it  be  required  to  multiply  both  terms  of  the  frac- 
tion I  by  4. 

Analysis. — In  |^,  the  fractional  unit  is  ^,  and  it  operation. 
is  taken  3  times.  By  multiplying  the  denominator  ■? "^  ^  ■_  12. 
by  4,  the  fractional  unit  becomes  X,  the  value  of  *  **  *  ^  ** 
which  is  \  times  as  great  as  \  By  multiplying  the  numerator 
by  4,  we  increase  the  number  ot  fractional  units  taken,  4  times; 
that  is,  we  increase  the  number  just  as  many  times  as  we  decrease 
the  1  Mite  ;  hence,  the  value  of  the  fraction  is  not  changed  ;  there- 
fore, we  liave 

rRorosiTiON  V. — If  both  terms  of  a  fraction  be  multiplied 
by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

EXAMPLES. 

1.  Multiply  the  numerator  and  denominator  of  |  by  7 : 
this  gives  y5|-H- 

2.  Multiply  the  numerator  and  denominator  of  ^  by  3,  by 
4,  by  5,  by  6,  by  9. 

3.  Multiply  each  term  of  Jf  by  2,  by  3,  by  4,  by  5,  by  G. 

162.  Let  it  be  required  to  divide  the  numerator  and  de- 
nominator of  ^  by  3. 

Analysis. — In  ^,  the  fractional  unit  is  ^,  and  operation. 
is  taken  6   times.     By  dividing  the   denominator        6  -t-3      2 


by  3,  the  fractional  unit  becomes  i.  the  value  of 


i5-:-3 


which  is  3  times  as  great  as  ^.  By  dividing  the 
numerator  by  3,  we  diminish  the  number  of  fractional  units  taken 
3  times  :  that  is,  we  diminish  the  number  just  as  many  times  as  we 
increase  itie  value:  hence,  the  value  of  the  fraction  is  not  changed : 
therefore,  we  have 

Proposition  VI. — Jf  both  terms  of  a  fraction  be  divided 
by  the  same  number,  the  value  of  the  fraction  vdll  not  be 
clmnged. 

EXAMPLES. 

1.  l)ivide  both  terms  of  the  fraction  xe  ^Y  ^  •  ^^  gives 

8    -^2_4 
Tg-2—6 


^^—^  Arts. 


ICI.  What  is  proved  in  proposition  V.  1 
1G2.    Wliat  is  proved  in  proposition  VI    \ 

c 


(51  KEDUCTTON   OF 

2.  Divide  both  terms  by  8  :  this  ^ves  ^^^:t|=:J. 

3.  Divide  both  terms  of  the  fraction  ^^  by  2,  by  4,  by  8, 
by  16. 

4.  Divide  both  terms  of  the  fraction  ^-^^  by  2,  by  3,  by  1, 
by  5,  by  6,  by  10,  by  12. 

REDUCTION  OF  FRACTIONS. 

163.  Keduction  of  Fractions  is  the  operation  of  changing 
the  fractional  unit  without  altering  the  value  of  the  fraction. 

A  fraction  is  in  its  lowest  terms,  when  the  numerator  and 
denominator  have  no  common  factor. 

CASE    I. 

164.    To  reduce  a  fraction  to  its  lowest  terms. 

1.  Reduce  ^-^  to  its  lowest  terms. 

Analysis. — By  inspection,  i1   is  seen  that  5 
is    a  common  lactor   of    the    numerator  and       1st.  operation, 
denominator.      Dividing  by   it,   we   have  |^.  5^ JLo___i4^ 

We  then  see  that  7  is  a  common  factor  of  14 
and   35  :   dividing   by   it,   we   have  f .     Now,  7  \  i  4  _  2 

there  is  no  factor  common  to  2  and  5  ;  hence,  ^^     ^' 

\  is  in  it.s  Imjoest  terms. 

The  areatest  common  divisor  of  70  and  175  2d  operation. 
is   35,   (Art.  136) ;  if  we  divide  both  terms  of  35)y'^=|. 

the  fraction  by  it,  we  obtain  ^.     The  value  of 
the  fraction  is  not  changed   in  either  operation,  since  the  numera- 
tor  and  denominator  are  both  divided  by  the  same  number  (Art. 
162)  :  hence,  the  following 

Rl'LE. — Divide  the  numerator  and  denominator  by  any 
number  thai  will  divide  them  both  without  a  remainder.,  and 
divide  the  quotient,  in  the  same  manner  until  they  have  no 
common  factor. 

Or  :  Divide  the  numerator  and  denominator  by  their  great 
est  common  diuisor. 

163  What  is  reduction  of  fractions  i  When  is  a  fraction  in  ite 
lowest  terms  T 

104.    How  ()o  you  reduce  a  iractiun  to  its  lowest  terms  '' 


COMMON    FKAOTIONB.  155      -^ 

EXAMPLES. 

Reduce  the  following  fractions  to  their  lowest  terms. 

1.  Reduce  |f .  :      9.  Reduce  ^|. 

2.  Reduce  if.  j    10.  Reduce  1^. 

3.  Reduce  fl. 

4.  Reduce  j^, 

5.  Reduce  f|. 

6.  Reduce  ^. 

7.  Reduce  ^5^. 

8.  Reduce  ^. 


11.  Reduce  if  1^. 

12.  Reduce  j^. 

13.  Reduce  ffj. 

14.  Reduce  T^nr- 

15.  Reduce  ff. 

16.  Reduce  fi£^. 


CASE  n. 

165.    To  redvce  an  improper  fraction  to  its    equivalent 
whole  or  mixed  nuwher. 

1.   In  *^  how  many  entire  units  ? 

Analysis. — Since  there  are  8  eighths  in  1  unit,  operation. 

in   '^  there  are  as  many  units  as  8  is  contain-  8)59 

ed  times  in  59,  which  is  7|  times.  7T 

Hence,  the  following 

Rule. — Divide  the  numerator  by  the  denominator,  and  the 

result  will  be  the  whole  or  mixed  number. 


examples. 

1.  Reduce  ^-^  and  y   to  their  equivalent  whole  or  mixed 
numbers. 

OPERATION.  OPERATION. 

4)84  9)67 

21  7J. 

2.  Reduce  ®^  to  a  whole  or  mixed  number. 

3.  In  ^-^  yards  of  cloth,  how  many  yards  1 

4.  In  ^^  of  bushels,  how  many  bushels  1 

105.    How  do  you  reduce  an  hup  roper  fraction  to  a  whole  or  iiiiied 
lunilier  *? 


156  KKDUOTION    OF 

5.  If  I  give  ^  of  an  apple  to  each  one  of  15  children,  how 
many  apples  do  I  give  1 

G.  Reduce  ffj,  \%\^,  s^^^  %3^7^2_s  ^  to  their  whole  or 
mixed  numbers. 

7.  If  I  distribute  878  quarter-apples  among  a  number  of 
boys,  how  many  whole  apples  do  I  use  ? 

8.  Reduce  %Vt¥.  tW'  'ttTO^'.  to  their  whole  or  mixea 
numbers. 

9.  Reduce  L±1R^.AJ^,  lA^^,  -^|-H^,  to  their  whole 
or  mixed  numbers. 

CASE    III. 

166.  To  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

1.  Reduce  4|  to  its  equivalent  improper  fraction. 

Analysis.  —  Since    in    any    number  operation 

there  are    5   times   as    many    fillhs    as  av'S  — 90  fflVia 

units,  in  4  there  will  be  5  times  4   fifths,  ai        a   I!f  i! 

or  20  fifths,  to  which  add  4  fifths,  and  .         ^^^     ^^  J   r®' 

we  have  24  fifths.  gives  Y   =24  filths. 

Hence,  the  following 

Rule. — Multiply  the  whole  number  by  the  denominator  of 
iKe  fraction  :  to  the  product  add  the  numerator^  and  place  the 
sum  over  the  given  denominator. 

EXAMPLES. 

1.  Reduce  47^  to  its  equivalent  fraction. 

2.  In  17|^  yards,  how  many  eighths  of  a  yard  1 

3.  In  42^^Q  rods,  how  many  twentieths  of  a  rod  1 

4.  Reduce  ^'^^^'^  to  an  improper  fraction. 

5.  How  many  112ths  in  20  5y*^? 

6.  In  84i^  days,  how  many  twenty-fourths  of  a  day  1 

7.  In  15^^  years,  how  many  365ths  of  a  year  1 

8.  Reduce  916|-§^  to  an  improper  fraction. 

9.  Reduce  25^,  156fJ,  to  their  equivalent  fractions. 


ir>G.  How  do  ycm  reduce  a  mixed  number  to  its  equivalent  inipropei 
fruttivin  1 


COMMON    FRA.CT10NS.  157 

CASE    VI. 

167.  To  reduce  a  whole  number  to  a  fraction  havmg  a 
given  denominator. 

1.  Eeduce  6  to  a  fraction  whose  denominator  shall  be  4. 

Analysis. — Since  in  1  unit  there  are  4  fourths.  operation. 
it  follows  that  in  6  units  there  are  6  times  4  fourths,  6  X  4  =  24, 
or  24  fourths:  therefore,  6=^  :  hence,  ^.. 

KuLE^ — Multiply  the  whole  number  and  denominator 
together,  and  write  the  product  aver  the  required  derimiii- 
nator. 

EXAMPLES. 

1.  Reduce  12  to  a  fraction  whose  denominator  shall  be  9. 

2.  Reduce  46  to  a  fraction  whose  denominator  shall  be  15. 


3.  Change  26  to  7ths. 

4.  Change  178  to  40ths. 

5.  Reduce  240  to  IHths. 


6.  Change  $54  to  quarters. 

7.  Change  96yc?.  to  quarters. 

8.  Change  426/6.  to  I6ths. 


CASE    V. 

168.    To  reduce  a  compound  fraction  to  a  simple  one. 

1.   What  is  the  value  jf  f  of  |^  1 

Analysis. — Three-fourths  of  ^  is  3  times  1  fourth  operation. 
of  |;  1  fourth  of  ^  IS  ^  (Art.  160) ;  3  fourths  of  f  is  3  X  5_  15 
3  times  ^,  or  ^  :  therefore,  |  of  4=M  •  hence,  ^  X  7  "28* 

Rule. — Multiply  the  numerators  together  for  a  new 
numerator,  and  the  denominators  together  for  a  new  de- 
nominator. 

Note. — If  there  are  mixed  numbers,  reduce  them  to  their  equiv 
alent  improper  fractions. 

EXAMPLES. 

Reduce  the  following  fractions  to  simple  ones. 

1.  Reduce  i  of  1  of  f .  j       4.  Reduce  2\  of  ^  of  7. 

2.  Reduce  f  of  |  of  f .  j       5.  Reduce  5  of  .J-  of  \  of  6. 
U    Reduce  {  of  I  of  ^j..       \       0.  Reduce  6 J  of  7^  of  6§  J 


158 


KEDUCTION    OF 


METHOD    BY    CANCELLING. 


169.  The  work  may  often  be  abridged  by  cancelling  com 
mon  Ikctors  in  the  numerator  and  denominator  (Art.  143). 

In  every  operation  in  fractions,  let  this  be  done  whenever 
it  is  possible. 


EXAMPLES. 

1 .  Reduce  f  of  |  of  f  to  a  simple  fraction. 


Here,         -x-X^  =  -, 


or, 


$    5 

7     0 
7     5 


Here. 


Note. — The  divisors  are  always  written  on  the  left  of  the 
vertical  line,  and  the  dividends  on  the  right. 

2.  Reduce  f  of  |  of  ^  to  its  simplest  temis. 

0     $      0      2 

Note. — Besides  cancelling  the  like  factors  8  and  8,  and  9  and  9, 
we  also  cancel  the  factor  3,  common  to  15  and  6,  and  write  over 
them,  and  at  the  right,  the  quotients  5  and  2. 

3.  Reduce  f  of  |  of  |  of  y^JL  of  ^  to  its  simplest  terms. 

4.  Reduce  j-\2_  of  _3_  of  _4^  of  ^  to  its  simplest  terms. 


5.   Reduce  3|  of  f  of 


Pit  of  49  to  its  simplest  terras. 


CASE   VI. 

170.  To  reduce  fractions  of  different  denominators  to 
fractions  having  a  common  denominator. 

1.  Reduce  ^,  \  and  |  to  a  common  denominator. 

167.  How  do  you  reduce  a  whole  number  to  a  fraction  having  a 
given  denominator  1 

IGy.  How  do  you  reduce  a  compound  fraction  to  a  simple  one. 

169.  How  is  the  reduction  of  compound  fractions  to  simple  ones 
aliriilyt'd  l»y  raru'cllati'in  'J 


i 


COMMuN    FRACTIONS. 


159 


0/»ERATTON 

1  x3x5—\5  1st  num. 
7x2x5  =  70  2d  num, 
4x3x2  =  24  3d  num. 
2x3x5  =  30  denom. 


Analysts. — If  both  terms  of  the 
first  fraction  be  multiplied  by  15, 
the  product  of  the  other  denomina- 
tors, it  will  become  ^^.  If  both 
terms  of  the  second  traction  be  mul- 
tiplied by  10,  the  product  of  the 
other  denominators,  it  will  become  ^  If  both  terms  of  the 
bird  be  multiplied  by  6,  the  product  of  the  other  denominators, 
t  will  become  ^.  In  each  case,  we  have  multiplied  both  terms 
of  the  fraction  by  the  same  number;  hence,  the  value  has  not 
been  altered  (Art.  161)  :  hence,  the  following 

Rule. — Redvce  to  simjile  fractimis  when  necessary  ;  then 
multiply  the  numerator  of  each  fraction  by  all  the  denomi- 
nators except  its  own^  for  the  new  numerators,  and  all  the 
denominators  together  for  a  cmnmon  denominator. 

Note. — When  the  numbers  are  small  the  work  may  be  per- 
formed mentally.     Thus. 

5>      4? 


20 
40' 


10 
40' 


13. 
40* 


EXAMPLES. 

Reduce  the  fbllowins:  fractions  to  common  denominators. 


1.  Reduce  |,  |,  and  \. 

2.  Reduce  f ,  y^y,  and  |. 

3.  Reduce  4,  i,  and  |. 

4.  Reduce  2^,  and  A  of  \. 

5.  Reduce  5^,  f  of  J,  and  4. 


6.  Reduce  3^  of  1  and  f. 

7.  Reduce  J,  ^,  and  37. 

8.  Reduce  4,  fi,  and  ^. 

9.  Reduce  7i  ^,  6f 
10.  Reduce  4^   ^,  and  2 J. 

Note. — We  may  often  shorten  the  work  by  multiplying  the  nu- 
merator and  denominator  of  each  fraction  by  such  a  number  as 
will  make  the  denominators  the  same  in  all. 

10.  Reduce  \  and  J  to  a  common  denominator. 

OPERATION. 

Analysis. — Multiply  both  terms  of  the  first  by  2=f 

3,  and  both  terras  of  the  second  by  2.  i  _2 

14.  Reduce  f,  3|,  and  f. 


11.  Reduce  \  and  ^. 


12    Reduce  i,  ^^,  and  ^. 
13.  Reduce  f ,  ^,  y\. 


15.  Reduce  6^,  9^,  and  5. 

16.  Reduce  7f,  J,  ^  and  J. 


170.  How  do  you  reduce  fractions  of  different  denominatora  to  frac- 
tious having  a  connnou  denominator  1  When  the  numbers  are  •small 
liuw  m;i}  llie  'Auik  lie  jjerforuiedl  ■•  , 


160  KEDUCnON   OF 

CASE   vn. 

171.   To  reduce  fractions  to  their  least  c&minon  denominator. 

The  least  common  denominator  is  the  number  which  con- 
tains only  the  prime  factors  of  the  denominators, 

I.  Reduce  J,  |,  and  f ,  to  their  least  common  denominator. 

OPERATION. 

(12-^3)  xl=   4   1st  Numerator.  3)3     .     6     .     4 

(12-^6)  x5:ii:10  2d         "  2)1     .     2     .     4 

(12-^4)  X  3=   9  3d         "  "~1     \     1     .     2 

3x2x2=12,  least  com.  denom. 

Therefore,  the  fractions  \,  J,  and  |-,  reduced   to  their  least 
nmon  denominator,  a 
Hence,  the  fbliowinsr 


common  denominator,  are  ^^,  \^,  and  ^ 


Rule. — I.  Find  the  least  common  dividend  of  the  denomu 
nators  (Art.  14:0),  which  will  be  the  least  common  denominator 
of  the  fractions. 

11.  Divide  the  least  common  denominator  by  the  denomina- 
f.ors  of  the  given  fractions  separately,  and  multiply  the  nume- 
rators  by  the  corresponding  quotients,  and  place  the  products 
over  the  least  common  denominator. 

Notes. — 1.  Before  beginning  the  operation,  reduce  every  frac- 
tion to  a  simple  fraction  and  to  its  lowest  terms. 

2.  The  expressions,  (12-^3)X1,  (12-t6)X5,  ( 1 2-^4) X 3,  indi- 
cate that  the  quotients  are  to  be  multiplied  by  1,  5,  and  3. 

EXA.MPLES. 

Reduce  the  following  fractions  to  their  least,  common 
denominator. 


2. 

Reduce  f ,  |,  /^. 

7. 

Reduce  31,  4^%  8^. 

3. 

Reduce  H-J,  6f,  5^. 

8. 

Reduce  1,  f ,  f ,  and  f. 

4. 

Reduce  fj,  ^,  f . 

9. 

Reduce2iofi  3iof2. 

5. 

Reduce  tVo.  4^.  f • 

10. 

Reduce  f ,  |,  f ,  and  ^V 

6. 

Reduce  fl,  3/^,  4. 

11. 

Reduce  I  f ,  f ,  1,  «• 

171.   What  is  the  least  common  denominator  of  several  fractious' 
How  do  vou  reduce  fractions  to  their  least  conuuoa  denominator  1 


COMMOM   FKA 


\DDITION  OF  FRA 

172.  Addition  of  Fractions  is  the  operation 
number  of  liactional  units  in  two  or  more  fractions. 

1.  What  is  the  sum  of  i,  |,  and  J  ? 

Analysis. — The  fractional  unit  is  the  same 
m  each  >  traction,  viz:  i;  but  the  numerators 
show  how  many  such  units  are  taken  (Art.  148)  ; 
hence,  the  svm  of  the  numerators  written  over 
the  common  denominator,  expresses  the  sutn  of 
the  fractions. 

2.  What  is  the  sum  of  i  and  |  ? 

Analysis. — In  the  first,  the  fractional  unit 
is  ^,  in  the  second  it  is  ^.  These  units,  not 
being  of  the  same  kind,  cannot  be  expressed  in 
the  same  colJection.     But   the  i=|,  and  §  =  f, 


OPERATION. 

1  +  3-1-0  =  9. 


Ans, 


4} 


OPERATION. 
1—3 
2—6 
2  _4 
3—^ 


+i=i=H 


in  each  of    which  the  unit    is  J  :  hence,   iheir 
sum  is  |~1^. 

Note. — Only  units  of  the  same  kind,  whether  fractional  or  inte- 
gral, can  be  expressed  in  the  same  collection. 

From  the  above  analysis,  we  have  the  following 

Rule. —  1.  When  the  fractions  have  the  same  denominator^ 
add  the  numerators,  and  place  the  sum  over  the  comTuon  deno- 
minator. 

II.  When  they  have  not  the  same  denominator,  reduce  them 
to  a  common  denominator,  and  then  add  as  before. 

Note. — After  the  addition  is  performed,  reduce  every  result  to 
its  lowest  terms. 


EXAMPLES. 


1.  Add  \,  I,  f ,  and  f 

2.  Add! 


13  6     oTifl 

^,    2.  ^.    ttl^tJ 

—    —  a  1  ul  — 

3      4  6      13 


^• 


3.  Addf,  i,  f,  V^andJ 

4.  Add  -/y,  t\,  t\,  and  ^ 

5.  Add  A,  f^,  and  ^\. 

6.  Add  1,  f .  1   and  ^\. 

7.  Add  I  f ,  |,  and  ^\. 

11 


8.  Add  f ,  f ,  1  and  ^^. 

9.  Add9,f,T-L|,andi. 
10".  Add  1,  f,  f ,  ±,  and  |. 


11.  Add  j%, 


f ,  ^6_,  and  f . 


12.  Add  1  |,  and  |. 

13.  Add  tV>  ?>  I  and  i 

14.  Add  t\,  f ,  i,  and  Z^. 


l(J2  SUIiTliAOTION    01'' 

15.  "WTiat  is  the  sum  oflQl,  6f,  and  4|  ? 

OPERATION. 

Whole  numbers.  Fractions. 

19  +  6+4  =  29  i+l+4^|4a=l^^. 

Sum  =  29  +  lT%V  =  30iV5- 

173.  Note. — When  there  are  mixed  numbers,  add  the   whole 
numbers  and  fractions  separately^  and  then  add  their  sums. 

Find  the  sums  of  the  following  fractions  : 

16.  Add  3i  Vy^o,  12A   II  .      ,  20.  Add  900yL   450|,  75if. 

17.  Add  16,  9f,  2oJ,  li.  :  21.   Addiof/-j-of  i^to^ofl. 

18.  Add  1  off,  I  of  9,  14^.    22.  Add  17|  to  |  of  .  7|. 

19.  Add  2^-85-,  6-1,  and  12if.       23.  Add  |,  7i   and  8f. 

24.  What  is  the  sum  off  of  12|  of  7|-,  and  |  of  25  ? 

25.  What  is  the  sum  of  3^  of  9f  and  ^j  of  32S|  1 

174.  1.  What  is  the  sum  of  ^  and  i? 

Note. — If  each   of  two  fractions  has  operation. 

1  for  a  numerator,  the  sum   of  the  frac-       1-4-1—6    15  — 11 
.,,    ,  '    1    .     xu  i-  ii     •  5    '6—30    I    30 — To* 

tions   will   be  equal  to  the  sum  of  their       1  1   i_5+6      n 
denominators  divided  by  their  product.         s  ""«— 5x6  ~30" 

2.  What  is  the  sum  of  ^  and  ^  1  of  J  and  ^  1 

3.  What  is  the  sum  of  j  and  Jg  ?  of  Jg  and  j\  ?  of  j\ 
and  1? 

4.  What  is  the  sum  of  J  and  y^j  "^  ^^  i  ^^^^  i  •  ^^  i 
and^V^ 

SUBTRACTION   OF   FRACTIONS. 

175.  Subtraction  of  Fractions  is  the  operation  of  finding 
the  difference  between  two  fractions. 

172.  What  is  addition  of  fractions?  When  the  fractional  unit  is  the 
same,  what  is  the  sum  of  the  fractions  !  What  units  may  he  exj)rt'ssed 
in  the  same  collection  ?     What  is  the  rule  for  the  addition  of  fractions  1 

173.  When  there  are  mixed  numbers,  how  do  you  add  ! 
1 74     When  two  fractions  have  1  for  a  numerator,  what  is  their  sum 

eqvml  to  ! 

175.   What  is  subtraction  of  fractious  1 


COMMON    FK ACTIONS.  11)3 

1.  "WTiat  is  the  difTerence  between  ^  and  1 1 

Analysis. — In  Ihis  example  the  fractional  unit  operation. 

is  i  :  there  are  5  such  units  in  tlie  minuend  and  | — |^^|=:i. 

3  in  the  .subtrahend  :  their  difference  is  2  eighths;  ^^^^.     i^ 

therefore.  2  is  written  over  the  common  denomi-  *    **' 
nator  8. 

2.  From  if  take  ^.  |       4.  From  ^ff  take\^. 

3.  From  f  take  f .  |       5.  From  ff|  take  -jff . 

6.  What  is  the  difTerence  between  J  and  J  1 

OPERATION. 

Analysis. — Reduce  both  to  the  same  frac-  5  _j  o. 

tional  unit  ^  :  then,  there  are  10  such  units  5^ V 

in  the  minuend   and   4    in  the  subtrahend :        ,  q       ^~  ^^ , 

tT""T2— T2— ^• 

A  lis,  1. 
From  the  above  analysis  we  have  the  following 

Rule. — I.  Wlien  the  fractions  have  the  same  denondnatw^ 
subtract  the  less  numeiator  from  the  yr eater,  and  place  the 
difference  over  the  common  denominator. 

II.  When  they  have  not  the  same  denominator,  reduce  them 
to  a  common  denominator^  and  then  subtract  as  before. 

EXAMPLES. 

Make  the  following  subtractions  : 


4.  From  1,  take  f^^. 

5.  Fromiof  12,  take  jf  of  J. 

6.  F'mf  ofllof7,takc'f  off. 


1.  From  f  take  f . 

2.  From  f  take  f . 

3.  From  /^  take  y\. 

7.  From  f  of  f  of  1  take  f^  off  of  1. 

8.  From  f  of  f  of  ^,  take  f  of  f  of  f 

9.  From  j\  of  fl  of  i   take  {^  off 

10.  What  is  the  difference  between  4 i  and  2^1 

OPERATION. 

21— 15   -  90  or,  01_9  6 


l5    _ 


2^  Aus.  2^5 


104  MULTII'LICATION   OF 

176.  Therefore  :  When  there  are  mixed  numbers,  chMugc 
both  to  improper  fractiois  and  subtract  as  in  Art.  175  ;  or, 
subtract  the  integral  and  /^-actional  numbers  separately^  and 
write  the  results. 

11.  From  84y7_  take  161        |     12.  From  246-|  take  164^. 

13.  From  7f  take  4J  :  f =^\  ^nd  i  =  2T- 

Note. — Since  we  cannot  take  ^  from  -^  we  operation. 
borrow  1,  or  |j,  from  the  minuend,  which  added       72 —76 
to  2j  =  fy  ;  then  -^  from  J^  leaves  J^.   We  must        41  —4 J 

now  carry  1  to  the  next  figure  of  the  subtrahend  ^^""73! 5" 

and  proceed  as  in  subtraction  of  simple  numbers.  ^^^^-   %r* 

14.  From  16f  take  5f.  |  16.  From  3Gf  take  27yV 

15.  From  26f  take  19^.  I  17.  From  400y5^  take  327|-. 

18.  From  i  take  ^j. 

Note. — When   the  numerators  are  1,  operation. 

the   difference    of   the  two    fractions   is  1.— ^1^  r^^l  — .^  =  .^ 

equal  to  the  differeuce  of  the  denomina-  j^ 1     11-8  _  _3_ 

tors  divided  by  their  product.  ^      TT  —  nxs  ~  ss 

19.  What   is  the   difference  between  J   and  \  ?     Between 
I  and  J5  ?   J  and  f,  ?  J^  and  3V  ?  yV  ^nd  ^V  ?  io  and  ^\^  ? 

MULTIPLICATION  OF  FRACTIONS. 

177.  MuLTirLTCATiON  of  Fractions  is  the  operation  of  taking 
one  number  as  many  times  as  there  are  units  in  another, 
when  one  of  the  numbers  is  fractional,  or  when  they  are  both 
fractional. 

1.  If  one  yard  of  cloth  cost  f  of  a  dollar,  what  will  4  yards 
cost? 

Analysis. — Four   yards  will  cost  4  operation. 

^imes  as  much   as    1    yard;  if  1    yard       5  x  4=^-^  — — =21. 
costs  5  eighths  of  a  dollar,  4  yards  will       ^  .  ^ 

cost  4  times  5  eighths  of  a  dollar,  which  are  20  eighths  :  there- 
fore, if  1  yard  cost  f  of  a  dollar,  4  yards  will  cost  ^  =  2^  dollars. 

17C.   When  there  arc  mixed  numbers,  how  do  you  subtract !  Explain 
ihc  case  when  the  fractional  part  of  the  subtrahend  is  the  greater  1 
177.    What  is  umltiplicatioi:  of  friicliouii  1 


COMMON    KBAOTIONS. 


165 


OPERATION. 

OR, 

$ 

2 

5 

: — 

Multiply  tl^ 

2d.  If  we  divide  the  denominator  by  4, 
the  fraction  will  be  multiplied  by  4  (Prop. 
11)  :  performing  the  operation,  we  obtain, 
I  which  =  2^  :  hence. 

To  multiply  a  fractior.  by  a  whole  number 
numerator^  or  divide  the  denominator  by  the  multiplier. 

EXAMPLES. 

1.  Multiply  ^  by  12.        14.  Multiply  ^V  ^Y  ^' 

2.  Multiply  li  by  7.  5.  Multiply  fff  by  49. 

3.  Multiply  Vs^  by  9.  |     6.  Multiply  ^  by  26. 

7.  If  1  dollar  will  buy  f  of  a  cord  of  wood,  how  much  will 
15  dollars  buy  ? 

8.  At  f  of  a  dollar  a  pound,  what  will    12  pounds  of  tea 
cost  ? 

9.  If  a  horse  eats  J  of  a  bushel  of  oats  in  a  day,  how  much 
will  18  horses  eat? 

10.  What  will  64  pounds  of  cheese  cost,  at  ^  of  a  dollar 
a  pound  ? 

11.  If  a  man  travel  |-  of  a  mile  an  hour,  how  far  will  he 
travel  in  16  hours  ? 

12.  At  f  of  a  cent  a  pound,  what  will  45  pounds  of  chalk 
costi 

13.  If  a  man  receive  ^^  of  a  dollar  for  1  day's  labor,  how 
much  will  he  receive  for  1 5  days  ? 

14.  If  a  family  consume  |^  of  a  barrel  of  flour  in  1  month, 
how  much  will  they  consume  in  9  months  ? 

15.  If  a   person  pays  \^  of  a  dollar  a  month  for  tobacco, 
how  much  does  he  pay  in  18  months'? 

181.    To  multiply  a  whole  number  by  a  fraction. 

1.  At  15  dollars  a  ton,  what  will  f  of  a  ton  of  hay  costl 

Analysis. — 1st.  Four-fifths  of   a   ton    will 
cost  4  times   as  much   as  1  fifth  of  a  ton ;  if  operation. 

1  ton  cost  15  dollars,  1  fifth  will  cost -I  of  15    (15-^5)  x  4rz  12. 
dollars,  or  3  dollars,  and  ^  will  cost  4  times  3 
dollars,  which  are  12  dollars. 

180.   How  do  ynu  multiply  a  frdctijn  by  a  whole  uumberl 


Iti(>  MULTU'LICATION    OK 


Or  :  2d.  4  fifths  of  a  ton  will   cost  1  fifth 
of  4  times  the  cost  of  1  ton;  4  times  15  is  60,       15x4-^5=12. 
and  1  fifth  of  60  is  12. 

.„   3 

Note. — Both  operations  may  he  combined 
in  one  by  the  use  of  the  vertical  line  and  can- 
cellation :  hence, 


12  Ana. 


Divide  the  whole  number  hy  the  denominator  of  the  fraction 
and  multiply  the  quotieiit  hy  the  numerator  ; 

Or  :  Multiply  the  whole  number  by  the  numerator  of  the 
fi'uction  and  divide  the  product  by  the  denominator'. 


EXAMPLES. 


1.  Multiply  24  by  J. 

2.  Multiply  42  by  jf 


3.  Multiply  105  byf. 
4    Multiply  64  by  i|. 


5.  What   is  the  cost   of  f  of  a  yard  of  cloth  at  8  dollars  a 
yaid  ? 

6.  If  an  acre  of  land  is  valued  at  75  dollars,  what  is  ^  of 
it  worth  ? 

7.  If  a  house  is  worth  320  dollars,  what  is  y^^  of  it  worth  ? 

8.  If  a  man  travel  46  miles  in  a  day,  how  far  does   he 
travel  in  |  of  a  day  ? 

9.  At    18  dollars  a  ton,  what  is  the  cost  of  y®^  of  a  ton  of 
hay  ? 

10.  If  a  man  earn  480  dollars  in  a  year,  how  much  does 
he  earn  in  ^  of  a  year  ? 

182.    To  multiply  one  fraction  by  another. 

1.  If  a  bushel  of  corn  cost  J  of  a  dollar,  what  will  f  of  a 
bushel  cost  1 

OPERATION. 

Analysis. — 5-sixths  of  a  bushel  will  cost      Jxf  =  ^|=:|-. 
I   times  as  much  as    1   bushel,   or  5  times  ^  | 

1  sixth  as  much  :  i  of  f  is  ^,   (Art.  180),  g         ^ 

and  5  times  ^  is  ijr=^|- :  hence,  0  |  5 

$  I  5=|. 

181.  How  Jo  you  multiply  a  whole  number  by  a  fraction  ? 


C<)MMUM   JTiACTlvWS. 


167 


Multiply  the  numerators  together  for  a  new  numerator  and 
the  deuoinmators  together  for  a  new  denominator. 

Notes. — 1.  When  the  multiplier  is  less  than  1,  we  do  not  take 
the  whole  of  the  multiplicand,  but  only  such  a  part  of  it  els  the 
multiplier  is  of  1. 

2.  When  the  multiplier  is  a  proper  fraction,  multiplication  does 
not  imply  increase,  as  in  the  multiplication  of  whole  numbers. 
The  product  is  the  same  part  of  the  multiplicand  which  the  multi- 
plier is  oM. 


EXAMPLES. 


Multiply  g 


^^yf 


2.  Multiply  T%  by 


3.  Find  the  pro't  off,  f,  ^. 

4.  Find  the  pro't  off,  ^,  ff 


5.  If  silk  is  worth  ^^  of  a  dollar  a  yard,  what  is  |  of  a  yard 
worth  ? 

6.  If  I  owTi  f  of  a  farm  and  sell  J  of  my  share,  what  part 
of  the  whole  farm  do  I  sell  ? 

7.  At  3^  of  a  dollar  a  pound,  what  will  ^^  of  a  pound  of 
tea  cost  ? 

8.  If  a  knife  cost  f  of  a  dollar  and  a  slate  f  as  much,  what 
does  the  slate  cost  ? 


9.  Multiply  51  by  \  of  f . 

Note. — Before  multiplying, 
reduce  both  fractions  to  the  form 
of  simple  fractions. 


—  2J. 


5J-^ 


OPERATION. 

;   1  ofl-^A. 


¥x 


5T 


5T- 

n 

1  . 


l  =  'LAns 


GENERAL    EXAMPLES. 


Mult.  lof|.of|by  ^. 
Mult.  T^o  by  i  ^^^' 


4.  Mult.  5  of  I  of  I  by  4f 

5.  Mult.   14  off  of  9  by  6f 

6.  Mult,  f  of  6  off  by  f  of  4. 


3.  Mult.  J  of  3  by  i  of  15^ 

183.    When  the  multiplicand  is  a  whole  and  the.  multi- 
plier  a  mixed  number. 


182.  How  do  you  multiply  one  fraction  by  another  1  When  the 
multiplier  is  less  than  1,  what  part  of  the  multiplicand  is  taken  T  If  the 
fraction  is  proper,  does  multiplication  imply  mcrease  1  What  part  is  the 
product  of  the  multiplicand  '? 


168  .  Division  of 

7.  What  is  the  prod^jfct  of  48  by  8^  1 

Note.- First  mtiltiply  48  by  i,  which  gives  4?*5YI!°^*q 
8  ;  then  by' 8,  which  gives  384,  and  the  sura j 392  ^o  S~qq^ 
is  the  product :  hence,  ^^  ^  °  — ♦^Q^ 

392 

Multiply  first  hy  the  fraction,  and  then  hy  the  whole 
number,  and  add  the  'products. 

8.  MuJt.  67  by  9^.  1      10.  Mult.   108  by  12f 

9.  Mult.   12f  by  9.  I      11.  Mult.  5f  by  3^. 

12.  What  is  the  product  of  6J,  2|  and  |  of  12. 

13.  What  will  24  yards  of  cloth  cost  at  3f  dollars  a  yard  ? 

14.  What  will  6|  bushels  of  wheat  cost  at  3|  dollars  a 
bushel  1 

15.  A  horse  eats  ^  of  J  of  12  tons  of  hay  in  three  months  ; 
how  much  did  he  consume  % 

16.  11  I  of  I  of  a  dollar  buy  a  bushel  of  corn,  what  will 
T^  of  y^  of  a  bushel  cost  % 

17.  What  is  the  cost  of  5 J  gallons  of  molasses  at  96 J  cents 
a  gallon  ? 

18.  What  will  7^  dozen  candles  cost  at  ^^  of  a  dollar  per 
dozen  ? 

19.  What  must  be  paid  for  175  barrels  of  flour  at  7f  dol- 
lars a  barrel  'I 

20.  If  I  of  f  of  2  yards  of  cloth  can  be  bought  for  one  dol 
lar,  how  much  can  be  bought  ibr  J  of  13  J  dollars  ? 

21.  What  is  the  cost  of  15f  cords  of  wood  at  3|  dollars  a 
cord? 

DIVISION  OF  FRACTIONS. 

184.  Division  of  Fractions  is  the  operation  of  finding  a 
number  which  multiplied  by  the  divisor  will  produce  the  divi- 
dend, when  one  or  both  of  the  parts  are  fractional. 

185,    To  divide  a  fraction  by  a  whole  number. 

1.  If  4  bushels  of  apples  cost  f  of  a  dollar,  what  will 
1  bushel  cost  1 

183.  How  may  you  mulfiply  when  the  muttipiicand  is  a  whole  and  tht 
multiplier  a  mixed  number  1 

184.  What  is  division  of  fractions  ! 

185.  How  do  you  divide  a  fraction  by  a  whole  number  1 


OOALMt)N    FKA 

Analysis. — Since  4  bushels  cost  f 
1  bushel  wiJl  cost  i  of  |  of  a  dollar, 
the  numerator  of  the  fraction  f  by  4 
f  (Art.  159). 


Multiplying  the  denominator  by  4  will  pro- 
duce the  same  result  (Art.  160)  :  hence, 


^4  =  5-«=§ 


Divide  the  numerator  or  multiply  the  denominator  hy  the 
divisor  S 


Note, — By  the  use  of  the  vertical  line  and  the 
principles  of  cancellation  (Art.  143),  all  operations 
in  division  of  fractions  may  be  greatly  abridged. 


9 


2=1 


EXAMPLES. 


1.  Divide  |f  by  6. 

2.  Divide  if  by  9. 

3.  Divide  W^  by  1 5. 

4.  Divide  Iffi  by  75. 


5.  Divide  jf  by  6. 
'    Divide  ^2_  by  12. 


6.  j^iviue  ^^  oy  lis. 

7.  Divide  Jf  by  20. 

8.  Divide  ^f  f  by  27. 


9.  If  6  horses  eat  ^^  of  a  ton  of  hay  in  1  month,  how  much 
will  one  horse  eat  ? 

10.  If  9  yards  of  ribbon  cost  f  of  a  dollar,  what  will  1  yard 
cost? 

11.  If  1   yard  of  cloth   cost  4  dollars,   how  much  can  be 
bought  for  f  of  a  dollar  ? 

12.  If  5  pounds  of  coffee  cost  ^  of  a  dollar,   what  will 
1  pound  cost '? 

13.  At  $6  a  barrel,  what  part  of  a  barrel  of  flour  can  be 
bought  for  I  of  a  dollar  \ 

14.  li    10    bushels   of  barley  cost    3 J  dollars,   what  will 
1  bushel  cost'^ 

Note. — We  reduce  the  mixed  number  to  •~>i._io 

an  improper  fraction   and  divide  as  in  the  3       3  * 

case  of  a  simple  fraction.  U)_l_i  n  —  i    A 

15.  If  21  pounds  of  raisins  cost   4J  dollars,  what   will  i 
pound  cost  ? 

16.  If   12  men  consume  6|  pounds  of  meat  in  a  day,  how 
much  does  1  man  consume  ? 


170  DI VISTON    i)¥ 

186.    To  divide  a  ivhole  nuinher  by  a  fraction. 

1.  At  |-  of  a  dollar  apiece,  how  many  hats  can  be  bought, 
for  6  dollars  % 

Analysis. — Since  \  of  a  dollar  will  operation. 

buy  one  hat.  6  dollars  will  buy  as  many  6-f-|-  =  6  X  5-f-4=:7i. 

hats  as  \  is  contained   times   in  6;  and  *             ^ 

as  there  are   5  times  as  many  fifths  as  i        o 

whole  things  in  any  number,   in  6  there  2  <C  ^ 

are  30  fifths,  and  4  fifths  is  contained  in  '  5 

30  fifths  7-^  times:  hence,  "sTTT 7i 

Invert  the  terms  of  the  divisor  and  multiply  (he  wltole  num- 
ber by  the  new  fraction. 

EXAMPLES. 

1.  Divide  14  by  f  j     3.  Divide  63  by  ^. 

2.  Divide  212  by  |^.  '     4.   Divide  420  by  ^j, 

5.  At  ]^  of  a  dollar  a  yard,  how  many  yards  of  cloth  can 
be  bought  tor  9  dollars? 

G.  li'  a  man  travel  -J  of  a  mile  in  1  hour,  how  long  will  it 
take  him  to  travel  10  miles  ? 

7.  If  I"  of  a  ton  of  hay  is  worth  9  dollars,  what  is  a  ton 
worth  I 

187.    To  divide  one  fractimi  by  another. 

1.  At  f  of  a  dollar  a  gallon,  how  much  molasses  can  bo 
bought  for  1  of  a  dollar  % 

Analysis. — Since   ^  of    a    dollar  operation. 

will  buy  1  gallon,  |  of  a  dollar  will      l-r-l:^!  x  ^^^\^^=.2^^. 
buy  as  many  gallons  as  |  is  contained  o  i  ^ 


2| 


times  in  i  :  one   is  contained  in  J,  i 

times  ;  but  \  is  contained  5  times  as 

many  times  as  1,  or  ^  times  ;  but  2  16  |  35=2^. 

fifths  is  contained  half  as  many  times 

as  I,  or  f  I  times,  equal  to  2^^  times :  hence, 

I.   Invert  the  terms  of  the  divisor. 

XL  Multiply  the  numerators  together  for  the  numerator 
of  the  quotient,  and  the  denominators  together  for  the  de- 
nominator of  the  quotient. 

18G.  How  Jo  you  divide  a  whole  number  by  a  fractiuu  "^ 


COMMON    Fit  ACTIONS.  171 

NoTUs. — 1.  If  the  vertical  line  is  used,  the  denominator  of  the 
dividend  and  the  numerator  of  the  divisor  fail  on  the  left,  and  the 
other  terms  on  the  right. 

2.  Cancel  all  common  factors. 

3.  If  the  dividend  and  divisor  have  a  common  denominator^ 
they  will  cancel,  and  the  quotient  of  their  numerators  will  be  the' 
answer. 

4.  When  the  dividend  or  divisor  contains  a  whole  or  mixed 
number,  or  compound  fractions,  reduce  them  to  the  form  of  simple 
fractions-  before  dividing. 

EXAMPLES. 

4.  Divide  f  of  J  by  ^^  of  11 


1.  Divide  y%  by  \ 

2.  Divide  j\  by  f 


3.  Divide  ^  by  ^. 


5.  Divide!  of  21  by  |  of  3|. 

6.  Divide  G^  by  21. 


7.  At  1  of  a  dollar  a  pound,   how  much   butter  can  be 
bought  for  11  of  a  dollar '{ 

8.  If  1  man   consume   \\  pounds  of  meat  in  a   day,  how 
many  men  would  8J  pounds  supply  ? 

9.  If  6  pounds  of  tea  cost  41  dollars,  what  does  it  cost  a 
pound  ? 

10.  At  1  of  a  dollar  a  basket,  how  many  baskets  of  peaches 
can  be  bought  for  111  dollars  ? 

11.  If  |-  of  a  ton  of  coal  cost  6f  dollars,  what  will  1  ton 
cost,  at  the  same  rate  ? 

12.  How  much  cheese  can  be  bought  for  1|  of  a  dollar  at 
1  of  a  dollar  a  pound  1 

13.  A  man  divided  21  dollars  among  his  children,  giving 
them  Y^^  of  a  dollar  a  piece  ;  how  many  children  had  he  1 

14.  How  many  times  will  \^  of  a  gallon  of  beer  fill  a  vessel 
holding  1  of  4  gallons  ? 

15.  How   many  times  is  1  of  J  of  27  contained  in  J  of  J 
ol  42f  ? 

16.  If  51  bushels  of  potatoes  cost  2f  dollars,  how  much  do 
they  cost  a  bushel  ? 

17.  If  John  can  w^alk  21  miles  in  1  of  a  day,  how  far  can 
he  walk  in  1  day  1 

18.  If  a  turkey  cost  1|-  dollars,  how  many  can  be  bought 
for  1 2|  dollars  \ 

19.  At  1  of  5^  of  a  dollar  a  yard,  how   many  yards  of  rib- 
bon can  be  bought  for  Jl  of  a  dollar  1 

187.   How  do  you  divide  one  fraction  by  another  ? 


172  KlCDUCTloN    OK 

REDUCTION  OF  COMPLEX  FRACTIONS. 
188.   Complex  Fractions  are  only  other  forms  of  expression 
for  the  division  of  fractions  :  thus ;  ±  is  the  same  as  4  divided 


by  J  ;  and  may  be  written,  ^  x  ^=^^=z2^. 

189.  To  reduce  a  complex  fraction  to  the  form  of  a  sim- 
ple fraction.. 

22 
1     Reduce     —5  to  its  simplest  form 

OPERATION. 
4 

4£-|  =¥-l-|x^=A  Ans.  ;  hence, 
3 

Rule. — Divide  the  numerator  of  the  complex  fraction  by  its 
denominator^ 

Or  :  Multiply  the  numerator  of  the  upper  fraction  into  the 
denominator  of  the  lower,  for  a  numerator  ;  and  the  denomi- 
nator of  the  vpper  fraction  into  the  numerator  of  the  loner,  for 
a  denominator. 

Notes. — 1.  When  either  of  the  terms  of  a  complex  fraction  is  a 
mixed  number,  or  compound  fraction,  it  must  first  be  reduced  to 
the  form  of  a  simple  fraction. 

2.  Wlien  the  vertical  line  is  used,  the  numerator  of  tha  ufper  and 
the  deiiotninator  of  the  lower  numbers  fall  on  the  right  of  the  verti- 
cal line,  and  the  other  terms  on  the  left. 

EXAMPLES. 

Reduce  the  following  complex  fractions  to  their  simplest  form  : 

25 


1.  Reduce  4 

2.  Reduce 


6i 


3.  Reduce  IJ? 

4.  Reduce  f  of    |. 

T 

0.  Reduce  I4I-. 


6.  Reduce 

7.  Reduce_ii J5_. 

1  of  15 

8.  Reduce  — ,  ■*-. 

21 

9.  Reduce  .li_. 

10.  Reduce  i^^3 
fJof46 


DEW O MIW A TK    Fit AOTlUNS. 


178 


DENOMINATE  FRACTIONS. 

190.  A  Denominate  Fraction  is  one  in  which  the  unit  of 
the  I'raction  is  a  denominate  number.  Thus,  f  of  a  yard  is  a 
denominate  fraction. 

191.  Reduction  of  denominate  fractions  is  the  operation 
of  changing  a  fraction  from  one  denominate  unit  to  another 
without  altering  its  value. 

There  are  four  cases  : 

1st.  To  change  from  a  greater  unit  to  a  less,  as  from  yards 
to  inches  : 

2d.  To  change  from  a  less  unit  to  a  greater : 

3d.  To  find  the  value  of  a  fraction  in  integers  of  lower 
denominations : 

4th.  To  find  the  value  of  integers  in  a  fraction  of  a  larger 
unit. 

These  cases  will  be  arranged  in  sets  of  two  and  two. 


192.  To  change  from  a 
greater  unit  to  a  less. 

1.  In  J  of  a  yard,  how 
many  inches] 


OPERATION. 

f  X3xl2=i-|ac=:20  inches. 

Analysis. — Since  in  1  yard 
there  are  3  teet,  in  f  yards  there 
are  |  times  3  t'eet=-^  feet.  And 
Bince  in  1  loot  there  are  12 
inches,  in  ^  feet  there  are  ^ 
times  12  inches=-l^=  20 inch's: 
hence, 

Rule. — Multiply  the  frac- 
tion and  the  products  ivhich 
arise  by  the  units  of  the  scale, 
in  succession  J  until  you  reach 
the  unit  required. 


193.  To  change  from  a 
less  unit  to  a  greater. 

1.  In  20  niches,  how  many 
yards  ? 


OPERATION. 


20  X 


T^ 


1  2  0 5   vnrfls 

s^  — F6  —  9  jaras. 


X4-  = 


Analysis. — Since  12  inches 
make  1  foot,  in  20  inches  there 
are  as  many  feel  as  12  inches  is 

•  contained  times  in  20  inches 
=  \^  feet;  and  as  3  feet  make 
1  yard,  in  ^^  feet  there  are  as 
many  yards  as  3  feet  is  contained 

;  times  in  ^  feet  =1^  =  ^  yards  : 

i  hence. 

Rule. — Divide  the  fraclioUy 
and  the  quotients  which  arise, 
by  tJie  units  of  the  scale,  in  suc- 
cession^ until  you  reach  the  unit 
required. 


188.   What  are  complex  fractions  1 

I8y.  How  do  you  reduce  complex  to  simule  fractious  ^ 


174 


DKJN OMIN  ATE    FK ACTION S. 


NoTi:. — It  will  be  found  most  convenient  in  fractions,  to  perfonn 
.the  operations  by  cancellation:  thus, 


5 
^  4 

20  inches. 


3 


5 


5= J  yards*. 


EXAMPLES. 

1.  Keduce  g^  of  a  hogshead  to  the  fraction  of  a  quart. 

2.  Reduce  g-^^  of"  a  bushel  to  the  fraction  of  a  pint. 

3.  Reduce  ^-gVo  ^^  ^  pound  Troy  to  the  fraction  of  a  grain 

4.  What  part  of  a  foot  is  j-^-q  °^  ^  furlong  ? 
o.  What  part  of  a  minute  is  ^^q  °^  ^  ^^Y  ^ 

6.  Reduce  ^  g^g'^q  q  of  a  cz^-/.  to  the  fraction  of  an  ounce. 

7.  Reduce  f  of  a  gallon  to  the  fraction  of  a  hogshead. 

8.  What  part  of  a  £  is  |  of  a  shilling  1 

9.  What  part  of  a  hogshead  is  |  of  a  quart  ? 

10.  What  part  of  a  mile  is  -^^  of  a  foot? 

1 1.  Reduce  ^^qq  of  <£  to  the  fraction  of  a  farthing. 

12.  Reduce  y^^  of  an  Ell  Eiig.  to  the  fraction  of  a  nail. 

13.  Reduce  f  of  a  nail  to  the  fraction  of  a  yard. 

14.  Reduce  ^  ol"^  of  a  foot  to  the  fraction  of  a  mile. 

15.  Reduce  ^yz  ^^  ^  ^^"  ^^  ^^^  fraction  of  a  pound. 

16.  Reduce  -J  of  3^-pt^^.  to  the  fraction  of  a  pound  Troy. 
1  /.  What  part  of  a  mile  is  J  of  a  rod  '? 

18.  What  part  of  an  ounce  is  ^^  of  a  scruple  '? 

19.  g^l^  of  a  day  is  what  portion  of  10  minutes? 

20.  What  part  of  ^  of  a  foot  is  j|^  of  a  furlong  1 
21..  Reduce 

pint. 


QTWo  °^  ^  hogshead  of  ale  to  the  fraction  of  a 


190.  What  is  a  denominate  fraction  1 

191     What  is  reduction  of  denominate  fractions  1     Huw  many  caecf 
are  there !     Name  them. 

192.   How  do  you  change  from  a  greater  unit  to  a  less  1 
19il.  How  do  you  chaiiiie  from  a  less  unh  to  a  greater  1 


DKNCJillNATE    FRACTIONS. 


176 


194.  To  find  the  value  of 
afractioji  in  integers  ofloiver 
denominations. 

I.  What  is  the  value  of  J 
of  a  pound  Troy  % 

Analysis. — |  of  a  pound  re- 
duced to  the  fraction  of  an  ounce 
is  I X 12  =  ^  of  an  ounce,  (Art. 
177.),  which  is  equal  to  9-| 
ounces  :  |  of  an  ounce  reduced 
to  the  fraction  of  a  pennyweight 
is  f  X  20=-^  of&pwt.,  or  \2pwt. 

OPERATION. 

Numer.     4 

12  oz.      pwt. 
Denom.       5)48(9  .  .  .  12 
45 
3 
20 

5)60" 
60 
Rule. — I.  Multiply  the 
numerator  of  the  fraction  by 
the  numher  which  will  re- 
duce It  to  the  next  loiver  de- 
nmnination  and  divide  the 
'product  by  the  denominator. 

II.  If  there  is  a  remain- 
der, reduce  it  in  the  same 
manner,  and  so  on,  till 
the  lowest  denomination  is 
obtained. 


195.  To  find  the  value  of 
integers  in  a  fraction  of  a 
higher  denommation. 

2.  Reduce  9o2.  X^jncts.  to 
the  fraction  of  a  pound  Troy. 

Analysis. — In  1  pound  there 
are  240  pennyweights  :  1  pen- 
nyweight is  -^^  of  a  pound  ;  and 
9  ounces  Vlfwis.  =  I92pwts.  is 
^llf  of  a  pound =^  of  a  pound. 


operation. 
1     lb.  oz.     pwts. 

12^  9  .  .    12 

12"  20_ 

20     Num.   192  _ 
240  Denora.  240"^ 


Rule. — I.  Reduce  the  given 
integers  to  the  lowest  de- 
nomination named,  and  the 
result  will  be  the  numerator 
of  the  required  fraction. 

II.  Reduce  1  unit  of  the 
required  denomination,  to  the 
denomination  of  the  numera- 
tm',  and  the  result  tvill  be 
the  denominator  of  the  re- 
quired fraction. 


examples. 

3.  What  is  the  value  of  J  of  a  tun  ol  wine  "? 

4.  What  part  of  a  tun  of  wine  is  Zhhd.  "^Igal.  2qt.l 


194.  How  do  you  find  the  value  of  a  fraction  in  integers  of  lower  de- 
nominations ! 

195.  How  do  you  find  the  value  of  integers  in  a  fraction  of  a  higher 
denoniination  ' 


176  ADDITION    AND    SUBTK ACTION    OF 

6,  What  is  the  value  of  ^^  of  a  yard  ? 

6.  What  is  the  value  off  of  a  month  ? 

7.  What  is  the  value  of  f  of  a  chaldron  ? 

8.  What  is  the  value  of  J  of  a  mile  ? 

9.  What  is  the  value  of  3^2  °^^  toni 

10.  What  is  the  value  of  f  of  3  days  ? 

11.  W'hat  is  the  value  of  ^  of  |^  o^^f  bushels  of  grain? 

12.  Reduce  Sgals.  2qts  to  the  fraction  of  a  hogshead. 

13.  Reduce  2fur.  o^rd.  2yd.  to  the  fraction  of  a  mile. 

14.  What  part  of  a  £  is  5s.  l\d,  ? 

15.  What  part  of  a  pound  Troy  is  1  Oo0.  \^pwt.  Qgr.  ? 

16.  Wc'vt.  Oqr.  \2lb.  loz.  lj(/r.  is  what  part  of  a  ton? 

17.  What  part  is  2pk.  Aqt.  of  \bu.  Zpk.  ? 

18.  24/6.  602.  is  what  part  of  3^r.  12/6.  \2ozA 

19.  Reduce  'iwk.  Id.  9h.  o6»i.  to  the  fraction  of  a  month  ? 

20.  Reduce  2R.  32rd.  Qyd.    to     the  fraction  of  an  acre. 

21.  Reduce  12s.  9c/.  IJ/crr.to  the  fraction  of  a  guinea. 

22.  What  is  the  value  of  j^ib.  apothecaries'  weight  1 

23.  What  part  of  an  Ell  Enghsh  is  S^-r.  2na.  \\in.  ? 

24.  What  is  the  value  of  ^hhd  ?. 

25.  What  is  the  value  off  of  3  barrels  of  beer? 

26.  What  is  the  value  of  ^^  of  a  cwt.  ? 

27.  Reduce  3°  15'  18J"  to  the  fraction  of  a  cign. 

28.  Reduce  3  J  inches  to  the  fraction  of  a  hand. 

29.  What  is  the  value  of  3L  of  a  hogshead  of  wine  ? 

30.  What  is  the  value  of  ^^  of  an  acre  of  land  ? 

ADDITION  AND  SUBTRACTION. 

196.   To  add  or  subtract  denominate  fractions, 
1.  Add  f  of  a  £  to  J  of  a  shilling. 

f  of  a  £  =  §  of  Y  =  V  o^  a  shilling. 
Then.  4/  +  |=2_4/  +  tI=¥¥  «•  =  ¥*'— ^45.  2d, 


196.  Give  the  rule  for  adding  and  subtracting  denominate  fractions. 


DENOMINATE    FRACTIONS.  177 

Or,  the  f  of  a  shilling  may  be  reduced  to  the  fraction  of  a  X  : 
thus, 

i  of  ^=Tfo  of  a  £=:-^  of  a  £  : 
then,  2_^J_^48_^^^5^ofa£, 

which  being  reduced,  gives  145.  2d.  Ans. 

2.  Add  f  of  a  year,  ^^  of  a  week,  and  ^  of  a  day. 

f  of  a  year=:f  of  ^^  days=31zi>^.  2cla. 
J  of  a  w*^ek=:J  of  7  days     =   -     -    2da,    8kr. 
J  of  a  Aay    =----=----    ohr. 
Ans.  Slwk.  Ada.  llAr. 

3.  From  -^  of  a  X  take  J  of  a  shilling. 

J  of  a  shillingz=i  of  j^  of  a  £=zA-  of  a  £. 
Then,  i-eV-M-^o-fS  of  a  £  =  9*.  8d. 

4.  From  If  Z6.  Troy  weight,  take  ^z. 

lb.  oz.  pwt.  gr. 

\\lh.^\lh.  of  yo2.  =  21o0.  =  l  9 

\oz.  =::iof  \o  ofW.  =  80^r.-0  0       3       8 

Ans.   1  8     16     16 

Rule. — Reduce  the  given  fractions  to  the  same  unity  and 
then  add  or  subtract  as  in  simple  fractions,  after  which  reduce 
to  integers  of  a  lower  denomination  : 

Or  :  Reduce  the  fractions  separately  to  integers  of  lower  de- 
nominations^ and  then  add  or  subtract  as  in  denominate  num- 
bers. 

EXAMPLES. 

5.  Add  \\  miles,  ^q  furlongs,  and  30  rods. 

6.  Add  f  of  a  yard,  f  of  a  foot,  and  |^  of  a  mile. 

7.  Add  f  of  a  cwt.,  ^^  of  a  /6.,  13o«.,  J  of  a  cwt.  and  6/6 

8.  From  J  of  a  day  take  f  of  a  second. 

9.  From  f  of  a  rod  take  f  of  an  inch. 

10.  From  y^g  of  a  hogshead  take  ^  of  a  quart. 

11.  From  \oz.  take  \'pwt. 

12.  From  ^cwt.  take  \^Jb. 

12 


178 


DDODECIMALS. 


13.  Mr.  Merchant  bought  of  farmer  Jones  22 J  bushels  of 
wheat  at  one  time,  19y^^  bushels  at  another,  and  3o^  at  an- 
other :  how  much  did  he  buy  in  all  1 

14.  Add  ^  of  a  ton  and  y^^  of  a  cwt. 

15.  Mr.  Warren  pursued  a  bear  for  three  successive  days; 
the  first  day  he  travelled  2«|  miles  ;  the  second  33^^  miles ; 
the  third  29 Jy  miles,  when  he  overtook  him  :  how  larhad  he 
travelled  1 

16.  Add  6^  days  and  52^^^  minutes. 

17.  Add  ^cwL,  8^lb.,  and  3j%lb. 

18.  A  tailor  bouirht  3  pieces  of  cloth,  containing  respect- 
ively, i8f  yards,  2 If  Ells  Flemish,  and  16|  Ells  English: 
how  many  yards  in  all  I 

19.  Bought  3  kinds  of  cloth  ;  the  first  contained  ^  of  3  of 
J  of  I  yards  ;  the  second,  j  of  J  of  5  yards  ;  and  the  third,  | 
of  I  of  ^  yards  :  how  much  in  them  all  ? 

20.  Add  l\cwt.  17f/6.  and  7joz. 

2 1 .  From  f  of  an  oz.  take  J  of  a  pwt. 

22.  Take  -^  of  a  day  and  ^  of  |  of  f  of  an  hour  from 
3|^  weeks. 

23.  A  man  is  GJ  miles  from  home,  and  travels  4mi.  I  fur. 
'^24rd, ,  when  he  is  overtaken  by  a  storm  :  how  far  is  he  then 

from  home  ? 

24.  A  man  sold  J^  of  his  farm  at  one  time,  ^  at  another, 
and  y2y  at  another  :  what  part  had  he  left  1 

25.  From  i^  of  a  £  take  f  of  a  shilUng, 

26.  From  Ijoz.  take  ^pwt, 

27.  From  8^cwL  take  ^^yh. 

28.  From  3^lb.  Troy  weight,  take  ^z. 

29.  From  1^  rods  take  f  of  an  inch. 

30.  From  f  f  fe  take  y^^  !  . 

DUODECIMALS. 

197.  If  the  unit  1  foot  be  divided  into  12  equal  parts,  each 
part  IS  called  an  inch  or  prune,  and  marked  '.  If  an  inch  be 
divided  into  12  equal  parts,  each  part  is  called  a  second,  and 
marked  ".      If  a  second  be  d-ividod.    in 


DU0UECIMAL6.  179 

equal  parts,  each  part  is  called  a  third,  and  marked  '"  ;  and 

so  on  for  divisions  still  smaller. 
This  division  of  the  foot  gives 

1 '  inch  or  prime  -  -  -  -  -  =  tV  ®^  ^  ^^^^' 
\"  second  is  j^  of  y'^  -  -  -  =  jj^-  of  a  foot. 
V"  third  is  yL  of  ^^2  of  ^  -     -     =T^  of  a  foot. 

Note. — The  marks  ',  ",  '",  &c.,  which  denote  the  fractionat 
itnitSy  are  called  indices. 

TABLE. 

12'"  make  1"  second. 

12"  "  1'  inch  or  prime. 

12'  "  1  foot. 

Hence  :  Duodecimals  are  denominate  fractions,  in  which 
the  primary  unit  is  I  foot,  and  12  the  acule  of  division. 

Note. — Duodecimals  are  chiefly  used  in  measuring  surfaces  and 
solith. 

ADDITION  AND  SUBTRACTION. 

198.  The  units  of  duodecimals  are  reduced,  added,  and 
subtracted,  like  those  of  other  denominate  numbers.  The 
8<ale  is  always  12. 

EXAMPLES. 

1.  In  185',  how  many  feet? 

2.  In  250",  how  many  feet  and  inches  ? 

3.  In  43G7'",  how  many  ieet  ? 

4.  What  is  the  sum  of  3//.  6'  3"  2'"  and  2ft.  1'  10"  11"'? 

5.  What  is  the  sum  of  6/7.  9'  7"  and  &fL  7'  3"  4'"? 

6.  What  is  the  difference  between  9/V.  3'  5"  6"'  and  Ift, 
3'  6"  7"'? 

7.  What  is  the  difference  between  40/if.  6'  6"  and  29/^  7'"  ? 

8.  What  is  the  difference  between  \2ft.  T  9"  6"'  and  4//-. 
9'  7"  9'"  % 

197.  If  1  foot  be  divided  into  twelve  equal  parts,  what  is  each  part 
called  1  If  the  inch  be  so  divided,  what  is  each  part  called  ?  Wha*.  are 
duodecimals  ?  ^  For  what  are  duodecimals  chiefly  used  ! 

1138.   Row  do  you  add  and  subtract  duodecimals  1   Wlmt  is  ihe  scale  1 


180  DUODECIMALS. 

MULTIPLICATION. 

199.  Begin  with  the  highest  unit  of  the  multiplier  and  the 
louest  of  the  multiplicand,  and  recollect, 

1st.  That  I  fbotX  1  ibotrz:l  square  foot  (Art.  110). 

2d.  That  a  part  of  a  foot  X  a  part  of  a  foot  =  some  part  of  a 
square  foot. 

Note. — Observe  that  the  unit  is  changed,  by  mulf.iplicalion, 
from  a  linear  to  a  superficial  unit. 

1.  Multiply  6ff.  1'  8"  by  2fL  9'. 

Analysis.  — Since  a  prime   is  ^  of  a 
foot  and  a  second  y^, 
2 X 8"  =^1^  of  a  square  foot;  which  re- 
duced to    12ths,  is   1'   and    4":  that  is, 

1  twelfth,  and  4  twelfths  of  :jlj  of  a  square 
foot. 

2  X7'  =14  twelfths  =:l/it.  2' 
2  X6    =12  square  feet, 
9'x8^'=yif5  of  a  square  foot=6" 
9'x7'==i6^=5'  3" 

9   X6'   =:f|=:4  6' 

Rule.- — I.  Write  the  multiplier  under  the  multiplicand^ 
so  that  units  of  the  same  order  shall  fall  in  the  same 
column. 

n.  Begin  ivith  the  highest  unit  of  the  multiplier  and 
the  lotcest  of  the  multiplicand,  and  make  the  index  of  each 
product  equal  to  the  sum  of  the  indices  of  the  factors. 

III.  Reduce  each  product,  in  succession,  to  square  feet^ 
and  \2ths  of  a  square  foot. 

Note. — The  index  of  the  unit  of  any  product  is  equal  to  the 
sum  of  the  indices  of  the  factors. 

EXAMPLES. 

1.  IIow  many  solid  feet  in  a  stick  of  timber  which  is  25 
feet  6  inches  long,  2  feet  7  inches  broad,  and  3  feet  3  inches 
thick  ? 


OPERaT[ON. 

'tr 

8" 

2  9' 

2x8"=       \' 

4// 

2x1'  =   I  2' 

2  X6   =12 

9'X8"  = 

6" 

9'x7'=       5' 

3" 

9'x6   =  4  6' 

Prod.     18  3' 

1" 

lUU.  Ejcpluhi  the   uicthud   of  iiiulti[ilyai{;   Juudccuuals.      Give   the 
rult-. 


DUODECIMALS. 


18i 


Beginning   with  the  2  feet,  we   say  2 


1   square  foot :  then,  2 
and    1  to   carry  are  51 


times  6'  are  12'  = 
times  25  are  50, 
square  feet. 

Next. 

hen  7' times  25=  175'=  14  7':  hence,  the 
surface  i&  65  10'  6",  and  by  multiplying 
by  the  thickness,  we  find  the  solid  contents 
to  be  214  1'  1"  6'"  cubic  feet. 


OPERATION. 

25     6'  lengtli. 

2     7'  breadth. 
51     0' 

3'  6" 
14     7' 


65   10'  6" 
3     3'  thicknesj* 


197     7'  6" 

16     5'  1"  ^"' 
214     1'  1"  6'" 


2.  Multiply  9/55.  4.m.  by  Sft.  Sin. 

3.  Multiply  9ft..  2in.  by  9fL  ein. 

4.  Multiply  24ft.  lOin.  by  6ft.  8m. 

5.  Multiply  70ft.  9m.  by  12/^;.  Sin. 

6.  How  many  cords  and  cord  feet  in  a  pile  of  wood  24  feet 
long,  4  feet  wide,  and  3  feet  6  inches  high  ? 

7.  How  many  square  feet  are  there  in  a  board  17  feet  6 
inches  in  length,  and  1  foot  7  inches  in  width  ? 

8.  "What  number  of  cubic  feet  are  there  in  a  granite  pillar 
3  feet  9  inches  in  width,  2  feet  3  inches  in  thickness,  and  12 
feet  6  inches  in  length? 

9.  There  is  a  certain  pile  of  wood,  measuring  24  feet  in 
length,  16  feet  9  inches  high,  and  12  feet  6  inches  in 
width.     How  many  cords  are  there  in  the  pile  ? 

10.  How  many  square  yards  in  the  walls  of  a  room,  14 
feet  8  inches  long,  11  feet  6  inches  wide,  and  7  feet  11  inches 
high  ? 

11.  If  a  load  of  wood  be  8  feet  long,  3  feet  9  inches  wide, 
and  6  feet  6  inches  high,  how  much  does  it  contain  ? 

1 2.  How  many  cubic  yards  of  earth  were  dug  from  a  cellar 
which  measured  42  feet  10  inches  long,  12  feet  6  inches  wide, 
and  8  feet  deep? 

13.  What  will  it  cost  to  plaster  a  room  20  feet  6'  long,  1«5 
feet  wide,  9  feet  6'  high,  at    18  cents  per  square  yard  ? 

14.  How  many  feet  of  boards  1  inch  tbick  can  be  cut  from 
a  plank  ]8ft.  9in.  long,  ]ft.  6in.  wide  itiid  Sin.  thick  if  there 
-M  uo  waete  in  sawing  i 


182  DECIMAL    FRACTIONS. 

DECIMAL    FRACTIONS. 

200.  There  are  two  kinds  ol'  Fractious  ;  Common  Frac- 
tions and  Decimal  Fractions. 

A  Common  Fraction  is  one  in  which  the  unit  is  divided 
into  any  number  of  equal  parts. 

A  Decimal  fraction  is  one  in  which  the  unit  is  divided  ac- 
cording to  the  scale  of  tens. 

201.  If  the  unit  1  be  divided  into  10  equal  parts,  the  parts 
are  called  tenths. 

If  the  unit  1  be  divided  into  one  hundred  equal  parts,  the 
parts  are  called  hundredths. 

If  the  unit  1  be  divided  into  one  thousand  equal  parts,  the 
parts  are  called  thousandths,  and  we  have  similar  expressions 
lor  the  parts,  when  the  unit  is  fiirther  divided  according  to  tbe 
scale  of  tens. 

These  fractions  may  be  written  thus  : 

Four-tenths,  -  "  "  '  iir* 


To- 

4  5 

125 
TOOO* 
1047 
10000" 

denominator 


Six-tenths,     -  -  -  - 

Forty-five  hundredths, 
125  thousandths, 
1047  ten  thousandths, 

From  which  we  see,  that  in  each   case  the 
indicates  the  fractional  unit ;  that  is,  determines  whether  the 
parts  are  tenths,  hundredths,  thousandths,  &c. 

202.  The  denominators  of  decimal  fractions  are  seldom 
set  down.  The  fractions  are  usually  expressed  by  means  of 
a  period,  placed  at  the  left  of  the  numerator. 

Thus,       ^  -         is  written   -         -         .4 


45 

ToTJ 


.45 
.12^ 
T^^'o -1047 


tWo .125 


200.  How  many  kinds  of  fractions  are  there  \  What  are  they  1 
What  IS  a  common  fraction  1  What  is  a  decimal  fraction  ] 

201.  When  the  unit  1  is  divided  into  10  equal  parts,  what  is  each 
part  called  ]  What  is  each  part  called  when  it  is  divided  into  100  equal 
parts  !  When  into  1000  !  Into  10,000,  &c.  1  How  are  decimal  frac- 
tiutis  ibriued  1     What  gives  denoiuinatioti  to  the  fractiun  1 


DKCTMAL    FI^ACTIONS.  ]  8S 

This  method  of  writing  decimal  fractions  is  a  mere  lan- 
guage, and  is  used  to  avoid  writing  the  denominators.  I'he 
denominator,  however,  of  every  decimal  fraction  is  always 
understood  : 

It  is  the  unit  1  u'ith  as  many  ciphers  annexed  as  there 
are  places  of  figures  in  the  decimal. 

The  place  next  to  the  decimal  point,  is  called  the  place 
of  tenths^  and  its  unit  is  1  tenth.  The  next  place.  Lo  the 
right,  is  the  place  of  hundredths,  and  its  unit  is  1  hundredth  ; 
the  next  is  the  place  ol"  thousandths,  and  its  unit  is  1  thous- 
andth ;   and  similarly  for  places  still  to  the  right. 

DECIMAL  NUMERATION  TABLE. 


« 

JS 


•    as    X    O        tJ 

.1  all-^'^ 


.4  is  read  4  tenths 

.5  4  -  -  54  hundredths. 

.0  6  4  -  -  64  thousandths. 

.6754  -  -  6754  ten  thousandths, 

.01234  -  -  1234  hundred  thousandths, 

.007654  -  -  7654  miliionths. 

.0043604  -  -  43604  ten  miliionths. 

Note. — Decimal    fraciions  are    numerated  from  left  ta  .  ;,^t' 
thus,  tenths,  hundredths,  thousandths,  &o. 


202.  Are  the  denominators  of  decimal  fractions  generally  set  down  1 
How  are  the  fractions  expressed?  Is  the  denominator  under? toed  I 
What  is  it  ?  What  is  the  place  next  the  decimal  point  called  !  V\hat 
is  its  unit]  What  is  the  next  place  called  ?  What  is  its  unit?  What 
is  the  third  place  called  !  What  is  its  unit  !  Which  way  are  deciiiials 
tiumeiated  I 


184  DECIMAL   FRACTIONS. 

203.  Write  and  numerate  the  following  decimals : 

Four-tenths,  -  -  .4 

Four  hundredths,  -  -  .0  4 

Four  thousandths,               -  -  .0  0  4 

Four  ten  thousandths,       -  -  .0004 

Four,  hundred  ihousandths,  '         -  .00004 

Four  milhonths,    -             -  -  .000004 

Four  ten  milHonths,          -  -  .0000004. 

Here  we  see,  that  the  same  figure  expresses  different  deci- 
mal units,  according  to  the  place  which  it  occupies  :  therefore. 

The  value  of  the  unit,  in  the  different  places,  in  passing 
from  the  left  to  the  right,  diminishes  according  to  the  scale 
of  tens. 

Hence,  ten  of  the  units  in  any  place,  are  equal  to  one  unit  in 
the  place  next  to  the  left  ;  that  is,  ten  thousandths  make  one 
hundredth,  ten  hundredths  make  one-tenth,  and  ten-tenths, 
the  unit  1. 

This  scale  of  increase,  from  the  right  hand  towards  the 
Left,  is  the  same  as  that  in  whole  numbers ;  therefore. 

Whole  numbers  and  decimal  fractions  may  be  united  by 
placing  the  decimal  point  between  them  :  thus, 

Whole  numbers.  Decimals. 


1  i 

re  s 

-  -  ^ 

i  § 


2  "^ 


5  -:     2     S     o 


3    -O    ^ 


c    2    3    c    5?    «    §    2    .^    §  :=; 


83630641. 0  478976 

A  number  competed  partly  of  a  whole  number  and  partly 
of  a  decimal,  is  called  a  mixed  number. 


DECIMAL   FUAOTIONS.  185 

RULE    FOR    WRITING    DECIMALS. 

Write  the  decimal  as  if  it  were  a  whole  numbei,  prefix- 
ing as  many  ciphers  as  are  necessary  to  make  it  of  tiie 
required  denomination. 

RULE    FOR    READING    DECIMAJ.S. 

Read  the  decimal  as  though  it  were  a  ivhole  number,^ 
adding  the  denomination  indicated  by  the  lowest  decimal 
unit, 

EXAMPLES. 

Write  the  following  numbers  decimally  : 

(1.)  (2.)  (3.)  (4.)  (5.) 

3  IG  17  32  165 


100  1000  10000  100  10000 

(6.)  (7)  (8)  (9)       V     (10.) 

18t^.         12t&5.         IGtMtj.         m%-         ll-W- 

Write  the  following  numbers  in  figures,  and  then  'aiimerat€ 
them. 

1.  Forty-one,  and  three-tenths. 

2.  Sixteen,  and  three  millionths. 

3.  Five,  and  nine  hundredths. 

4.  Sixty-five,  and  fifteen  thousandths. 

5.  Eighty,  and  three  millionths. 

6.  Two,  and  three  hundred  millionths 

7.  Four  hundred,  and  ninety -two  thousandths. 

8.  Three  thousand,  and  twenty-one  ten  thousandths, 

9.  Forty-seven,  and  twenty -one  hundred  thousandths, 

10.  Fifteen  hundred,  and  three  millionths. 

1 1 .  Thirty-nine,  and  six  hundred  and  forty  thousandths. 

12.  Three  thousand,  eight  hundred  and  forty  millionths, 

13.  Six  hundred  and  fifty  thousandths. 

803.  Does  the  value  of  the  unit  of  a  figure  depend  upon  the  place 
which  it  occupies  \  How  does  the  value  change  from  the  left  towards 
the  right  \  What  do  ten  units  of  any  one  place  make '  How  do  the 
units  of  the  places  increase  from  the  right  towards  the  left  ?  How  may 
whole  numbers  be  joined  with  decimals  1  What  is  such  a  number 
called  \  Give  the  rule  for  writing  decuiial  fractions.  Give  the  rule 
fur  reading  decimal  fniclions. 

7 


186  UNITED    STA'fEB    MONEY. 

UNITED  STATES  MONEY. 

204.  The  denominations  of  United  States  Money  correspond 
to  the  decimal  division,  if  we  regard  1  dollar  as  the  unit. 

For,  the  dimes  are  tenths  of  the  dollar^  the  cents  are  hiiU' 
dredths  of  the  dollar^  and  the  mills^  being  tenths  of  ike  cent, 
are  thousandths  of  the  dollar . 

EXAMPLES. 

1.  Express  $39  and  39  cents  and  7  mills,  decimally. 

2.  Express  $12  and  3  mills,  decimally. 

3.  Express  $147  and  4  cents,  decimally. 

4.  Express  $148  4  mills,  decimally. 

5.  Express  $4  6  mills,  decimally. 

6.  Express  $9  6  cents  9  mills,  decimally. 

7.  Express  $10  13  cents  2  mills,  decimally. 

AI^NEXING  AND  PREFIXING  CIPHERS. 

205.  Annexing  a  cipher  is  placing  it  on  the  right  of  a 
number. 

If  a  cipher  is  annexed  to  a  decimal  it  makes  one  more  deci- 
mal place,  and  therefore,  a  cipher  must  also  be  added  to  tlie 
denominator  (Art.  202). 

The  numerator  and  denominator  will  therefore  have  been 
multiplied  by  the  same  number,  and  consequently  the  value 
of  the  fraction  will  not  be  changed  (Art.  161) :  hence, 

Annexing  ciphers  to  a  decitnal  fraction  does  not  alter  iU 
value. 

We  may  take  as  an  example,  .3=^. 

If  we  annex  a  cipher  to  the  numerator,  we  must,  at  tho 
same  time,  annex  one  to  the  denominator,  which  gives, 


204.  If  the  denominations  of  Federal  Money  be  expressed  decimally, 
what  is  the  unit  1  What  part  of  a  dollar  is  1  dime  1  What  part  of  a 
dime  is  a  cent  1  What  part  of  a  cent  is  a  mill  1  What  part  of  a  do\[<xt 
is  1  cent  1     1  mill  1 

20.5.  When  is  a  cipher  annexed  to  a  number  1  Does  the  annexing 
of  ciphers  to  a  decimal  alter  its  value  1  Why  not  1  What  does  three 
tenths  become  by  annexing  a  cipher  1  W^hat  by  annexing  two  ciphers  1 
Three  ciphers  1  Wliat  does  8  tenths  become  by  annexing  a  cipher  ^  By 
aiinexh)g  two  ciphers  ?     IJy  ainicxing  three  ciphers'? 


DECIMAL    FK ACTIONS.  187 

.3  =     YoTS     =  .30       by  annexing  one  cipher, 
.3  =    YoipQ     =  .300     by  annexing  two  ciphers, 
.3  =   /^^^   ~  .3000  by  annexing  three  ciphers. 
Also,  ,5=j%--=.50=j'^  =  .600-^j^-^. 

Also,  .8  =  .80:=r.800  =  .8000=::.80000. 

206.  Prefixing  a  cipher  is  placing  it  on  the  left  of  a 
punibei'. 

If  ciphers  are  prefixed  to  the  numerator  of  a  decimal  frac- 
tion, the  same  number  of  ciphers  must  be  annexed  to  the 
denominator.  Now,  the  numerator  will  remain  unchanged 
while  the  denominator  will  be  increased  ten  times  for  every 
cipher  annexed  ;  and  hence,  the  value  of  the  fraction  will  be 
dcminished  ten  times  for  every  cipher  prefixed  to  the  nume- 
rator (Art.  160). 

Prefixing  ciphers  to  a  decimal  ft  action  diminishes  its 
value  ten  times  for  every  cipher  prefixed. 

Take,  for  example,  the  fraction  ,2=z^^. 
.2  becomes     ^^     =  .02       by  prefixing  one  cipher, 
.2  becomes    yo°^     =  '^^'^     ^Y  Prefixing  two  ciphers, 
.2  becomes  yoooo   —  -00 O'^  ^Y  prefixing  three  ciphers  : 

in  which  the  fraction  is  diminished  ten  times  for  every  cipher 

prefixed. 

ADDITION  OF  DECIMALS. 

207.  It  must  be  remembered,  that  only  units  of  the  same 
kind  can  be  added  together.  Therefore,  in  setting  down 
decimal  pumbers  for  addition,  figures  expressing  the  same 
unit  must  be  placed  in  the  same  column. 

206.  When  is  a  cipher  prefixed  to  a  number  1  When  prefixed  to  a 
decimal,  does  it  increase  the  numerator  ]  Does  it  increase  the  denomi- 
nator i  "What  effect  then  has  it  on  the  value  of  the  fraction?  What 
do  .2  become  by  prefixing  a  cipher  ?  By  prefixing  two  ciphers  ?  By 
prefixing  three  ]  What  do  .07  become  by  prefixing  a  cipher  I  By  pre- 
fixing two  ?     By  prefixing  three  1     By  prefixing  four  ? 

207.  What  parts  of  unity  may  be  added  together  1.  How  do  y^u  set 
down  the  numbers  for  addition  ?  How  will  the  decimal  points  fall  \ 
How  do  you  then  add  1  How  many  decimal  places  du  you  point  uiT  in 
lilt'  ?uin  ' 


188  ADi)rru)N  oir^ 

The  addition  of  decimals  is  then  made  in  the  same  inannei 
as  that  of  whole  numbers; 

1.  Find  the  sum  of  37.04,  704.3,  and  .0376. 

OPERATION. 

Place  the  decimal  points  in  the  same  column:  37.04 

this   brings  units  of  the  same  value  in   the   same        704.3 
column  :  then  add  as  in  whole  numbers  :  hence,  .0376 


741.3776 
Rule. — I.   Set  down  the  numbers  to  be   added  so  that 

figures  of  the  same  unit  value  shall    stand  in  the  same 

coluTnn. 

II.   Add  as  in  simple  numbers^  and  point  off  in  the  sum^ 

from  the  right  hand,  as  many  places  for  decimals  as  are  equal 

to  the  greatest  number  of  places  in  any  of  the  numbers  added. 

Proof. — The  same  as  'n  simple  numbers. 

EXAMPLES. 

1.  Add  4.035,  763.196,  445.3741,  and  91.3754  together. 

2.  Add  365.103113,  .76012,  1.34976,  .3549,  and  61.11 
together. 

3.  67.407  +  97.004-f  44-.6  +  .06  +  .3 

4.  .0007+1.0436-f  .4  +  .05-f-.047 

5.  .0049  +  47.04264-37.0410-1-360.0039  =  444.0924. 

6.  What  is  the  sum  of  27,  14,  49,  126,  999,  .469,  and 
.2614? 

7.  Add  15,  100,  67,  1,  5,  33,  .467,  and  24.6  together. 

8.  What  is  the  sum  of  99,  99,  31,  .25,  60.102,  .29,  and 
100.347  \ 

9.  Add  together  .7509,  .0074,  69.8408,  and  .6109. 

10.  Required  the  sum  of  twenty-nine  and  3  tenths,  four 
hundred  and  sixty-five,  and  two  hundred  and  twenty-one 
thousandths, 

11.  Required  the  sum  of  two  hundred  dollars  one  dime 
three  cents  and  9  mills,  four  hundred  and  forty  dollars  nine 
mills^  and  one  dollar  one  dime  and  one  mill. 

12.  What  is  the  sum  of  one-tenth,  one  hundredth,  and  one? 
thousaudth  ? 


DKCIMAL 

13.  What  is  the  sum  of  4, 

14.  Required,  in  dolJars  and  del 
one  dime  one   cent  one  mill,  six 
lars  eight  cents,  nine  dollars  six  mills,  ort^*feiadEed-4oilars  six 
dimes,  nine  dimes  one  mill,  and  eight  dollars  six  cents. 

15.  What  is  the  sum  of  4  dollars  6  cents,  9  dollars  3  mills, 
14  dollars  3  dimes  9  cents  1  mill,  104  dollars  9  dimes  9  cents 
9  mills,   999  dollars  9  dimes  1  mill,  4   mills,  6   mills,  and   1 

miirr 

16.  If  you  sell  one  piece  of  cloth  for  $4,25,  another  for 
85,075,  and  another  for  $7,0025,  how  much  do  you  get  for 
all? 

17.  What  is  the  amount  of  $151,7,  $70,602,  $4,06,  and 
$807,2659] 

18.  A  man  received  at  one  time  $13,25  ;  at  another  $8,4  ; 
at  another  $23,051  ;  at  another  $6  ;  and  at  another  $0,75  : 
how  much  did  he  receive  in  all  1 

19.  Find  the  sum  of  twenty-five  hundredths,  three  hundred 
and  sixty-five  thousandths,  six  tenths,  and  nine  millionths. 

20.  What  is  the  sum  of  twenty-three  millions  and  ten,  one 
thousand,  four  hundred  thousandths,  twenty-seven,  nineteen 
millionths,  seven  and  five  tenths  ? 

21.  What  is  the  sum  of  six  millionths,  four  ten-thousandths, 
19  hundred  thousandths,  sixteen  hundredths,  and  four  tenths  1 

22.  If  a  piece  of  cloth  cost  four  dollars  and  six  mills,  eight 
pounds  of  cofiee  twenty-six  cents,  and  a  piece  of  muslin  three 
dollars  seven  dimes  and  twelve  mills,  what  will  be  the  cost 
of  them  all  ? 

23.  If  a  yoke  of  oxen  cost  one  hundred  dollars  nine  dimes 
and  nine  mills,  a  pair  of  horses  two  hundred  and  fifty  dollars 
five  dimes  and  fifteen  mills,  and  a  sleigh  sixty-five  dollars 
eleven  dimes  and  thirty-nine  mills,  what  will  be  their  entire 
cost  ? 

24.  Find  the  sum  of  the  following  numbers  :  Sixty-nine 
thousand  and  sixty-nine  thousandths,  forty-seven  hundred  and 
forty-seven  thousandths,  eighty-five  and  eighty-five  hun- 
dredths, six  hundred  and  forty-nine  and  six  hundred  aii'* 
forty-nine  ten-thousandths  ? 


190  SUBTKAUTION    OF 


SUBTRACTION  OF  DECIMALS. 

208     Subtraction  of  Decimal  Fractions  is  the  operation  of 
finding  the  difference  between  two  decimal  numbers. 

I.  From  3.275  to  take  .0879. 

NorB.  —  111  this   example  a  cipher    is    annexed  operation 

to  tlie  minuend   to   make   the  number  of  decimal  3.2750 

places  equal  to  the  number  in  the  subtrahend.  This  .0879 

does  not  alter  the  value  of  the  minuend  (Art.  205)  :  lS~^'on7 

hence,  ^-^^^^ 

Rule. — I.    Write  the  less  number  under  the  greater^  so  that 
figures  of  the  same  unit  value  shall  fall  in  the  same  column. 

II.  Subtract  as  in  simple  numbers^  and  point  off  the  deci- 
mal j)  laces  in  the  remainder,  as  in  addition. 

Proof. — Same  as  in  simple  numbers. 

EXAMPLES. 

1.  From  3295  take   0879. 

2.  From  291.10001  take  41.375. 

3.  From  10.000001  take  .111111. 

4.  From  396  take  8  ten-thousandths. 

5.  From  1  take  one  thousandth. 

6.  From  6378  take  one-tenth. 

7.  From  365.0075  take  3  millionths. 

8.  From  21.004  take  97  ten-thousandths. 

9.  From  260.470^  take  47  ten-millionths. 

10.  From  10.0302  take  19  miUionths. 

11.  From  2.01  take  6  ten-thousandths. 

12.  From  thirtv-five  thousands  take  thirty-five  thousandths. 

15.  From  4262.0246  take  23.41653. 

14.  From  346.523120  take  219.691245943. 

16.  From  64.075  take  .195326. 

lb.  What  is  the  difierence  between  107  and  .0007  1 

17.  What  is  the  difierence  between  \.d  and  .3735  ? 
lb.  From  96.71  take  96.709. 


208.  What  is  subtraction  of  decimal  fractions  1  How  do  you  set  down 
the  numbers  for  subtraction  ?  How  do  you  then  suljlnict  ^  How  many 
Jeriuial  y\xiVM»  d.)  yuu  point  uOin  i\it  reuir-indtr  1 


DECIMAL    FKACTTONS.  191 

MULTIPLICA.TION  OF  DECIMAL  FRACTIONS. 
209.   To  multij)ly  one  decimal  by  another. 
1.  Multiply  3.05  by  4.102. 

OPERATION. 

Analysis. — If  we  change  both  factors  to  vul-  Sjljis,— 3.05 

gar  fractions,  the  product  of  the  numerator  will  4  i  o  2  _. , j^  ino 

be  the  same  as  that  of  the  decimal  numbers,  and  Tooo  —  _i 

the  number  of  decimal  places  will  be  equal  to  the  610 

number   of    ciphers   in   the   two  denominators :  305 

hence,  12  20 

12.51110 

Rule. — Multiply  as  in  simple  numbers^  and  point  off  in 
the  product,  from  the  right  hand,  as  many  figures  for  decimals 
as  there  are  decimal  places  in  both  factors ;  and  if  there  be 
not  so  many  in  the  product,  supply  the  dejiciency  by  prefixing 
ciphers. 

EXAMPLES. 

1.  Multiply  3.049  by  .012. 

2.  Multiply  365.491  by  .001. 

3.  Multiply  496.0135  by  1.496. 

4.  Multiply  one  and  one  millionth  by  one  thousandth. 

5.  Multiply  one  hundred  and  forty-seven  millionths  by  one 
millionth, 

6.  Multiply  three  hundred,  and  twenty-seven  hundredths 
by  31. 

7.  Multiply  31.00467  by  10,03962. 

8.  What  is  the  product  of  five-tenths  by  five-tenths  ? 

9.  What  is  the  product  of  five-tenths  by  five-thousandths  % 

10.  Multiply  596.04  by  0.00004. 

11.  Multiply  38049.079  by  0.00008. 

12  What  will  6.29  weeks'  board  come  to  at  2,75  dollars 
per  week  ? 

13.  What  will  61  pounds  of  sugar  come  to  at  $0,234  per 
pound  \ 

209.  After  multiplying,  how  many  decimal  places  will  you  point  off 
in  the  product  \  When  there  are  not  so  many  in  the  product,  what  d* 
yoL'  'lol     Give  the  rule  for  tiio  multiplication  of  dec i tads. 


192 


CONTKA0TION6. 


14.  If  12.836  dollars  are  paid  for  one  barrel  of  flour,  what 
mil  .354  barrels  cost  1 

15.  What  are  the  contents  of  a  board,  .06  feet  long  and  .06 
wide  ? 

16.  Multiply  49000  by  .0049. 

17.  Bought  1234  oranges  for  4.6  cents  apiece  :  how  much 
lid  they  costi 

18.  What  will  375.6  pounds  of  coffee  cost  at  .125  dollars 
per  pound  ? 

19.  If  I  buy  36.'251  pounds  of  indigo  at  $0,029  per  pound, 
what  will  it  come  to  ? 

20.  Multiply  $89.3421001  by  .0000028. 

21.  Multiply  $341.45  by  .007. 

22.  What  are  the  contents  of  a  lot  which  is  .004  miles  long 
aiid   004  miles  wide"? 

23.  Multiply  .007853  by  .035. 

24.  What  is  the  product  of  $26.000375  multiplied  by 
.00007  ? 

CONTRACTIONS. 

210.  When  a  decimal  number  is  to  be  multiplied  by  10, 
100,  1000,  &c.,  the  multiplication  may  be  made  by  removing 
the  decimal  point  as  many  places  to  the  right  hand  as  there 
are  ciphers  in  the  multiplier,  and  if  there  be  not  so  many 
figures  on  the  right  of  the  decimal  point,  supply  the  deficieiicy 
by  annexing  ciphers. 


Thus,  6.79  multipUed  by 


Also,  370.036  multiplied  by 


210.  How  do  you  multiply  a  decimal  number  by  10,  100,  1000,  Ac.  1 
If  there  are  not  as  many  decimal  figures  as  there  are  cijihers  in  tue 
multiplier,  what  do  you  do  I 


10 

1 

r    67.9 

100 

679. 

^  1000 

►  =  ' 

6790. 

10000 

67900. 

100000^ 

679000. 

10    1 

3700.36 

100 

37003.6 

1000 

>  =  < 

370036. 

10000 

3700360. 

100000 

37003600. 

DECIMAL    FRACTIONS. 


DIVISION  OF  DECIMAL  FRACTIONS. 


193 


211.  Division  of  Decimal  Fractions  is  similar  to  that  of 
limple  numbers. 

1.   Let  it  be  required  to  divide  L38483  by  60.21. 

Analysis. — The  dividend  must  be  equal  operation. 

the  product  of  the  divisor  and  quotient,  60.21)1.38483(23 
(Art.    61);  and    hence  must   contain    as  1.2042 

many  decimal  places  as    both  of  them  ;  ISOfS" 

therefore, 

There  must  be  as  many  decimal  places  in  

the  qiLotient  as  the  decimal  places  in  the  divi-  Ans.  .023 

dend  exceed  those  in  the  divisor  :  hence, 

Rule. — Divide  as  in  simple  numbers,  and  point  off  in  the 
quotient,  from  the  right  hand,  as  many  places  for  dechnals  as 
the  decimal  places  in  the  dividend  exceed  those  in  the  divisor  ; 
and  if  there  are  not  so  many,  supply  the  deficiency  by  pre  fir- 
ing ciphers. 

EXAMPLES. 


1.  Divide  2.3421  by  2.11. 

2.  Divide  12.82561  by  3.01. 

3.  Divide  33.66431  by  1.01. 


4.  Divide  .010001  by  .01. 

5.  Divide  8.2470  by  .002. 

6.  Divide  94.0056  by  .08. 


7.  What  is  the  quotient  of  37.57602,  divided  bv  3  ;  by  .3  , 
by  .03  ;  by  .003  ;  by  .0003  ? 

8.  What  is  the  quotient  of  129  75896,  divided  by  8  ;  by 
.08  ;  by  .008  ;  by  .0008  ;  by  .00008  ? 

9.  What  is  the  quotient  of  187.29900,  divided  by  9 ;  by 
.9  ;  by  .09  ;  by  .009  ;   by  .0009  ;  by  .00009  ? 

10.  What  is  the  quotient  of  764.2043244,  divided  by  6  ; 
by  .06  ;  by  .006  ;  by  .0006  ;  by  .00006  ;  by  .000006? 

Note. — 1.  When  there  are  more  decimal  places  in  the  divisor 
than  in  the  dividend,  annex  ciphers  to  the  dividend  and  n)ake  the 
decimal  places  equal ;  all  the  figures  of  the  quotient  will  then  be 
whole  nur^jpers. 

211.  How  does  the  numhor  of  ilecimal  places  in  the  dividend  coro- 
pare  with  thai  in  the  divi.sor  and  quotient  !  How  do  you  determine 
the  number  of  df'cimal  places  in  the  quotient  {  If  the  divisor  contains 
four  places  and  the  dividend  six,  how  many  in  the  quotient  \  If  the 
divisor  contains  three  places  and  the  dividend  five,  how  many  in  the 
quotient !     CJive  the  rule  for  the  division  of  decimals. 

13 


lU 


DIVISION    OF 


EXAMPLES. 


1    Divide  4397.4  by  3.49. 


Note. — We  annex  one  0  to 
tlie  dividend.  Had  it  contained 
no  decimal  place  we  should 
have  annexed  two. 


OPERATION. 

3.49)4397.40(1260 
349 

907 
698 


2094 
2094 


Ans.   1260. 

2.  Divide  2194.02194  by  .100001. 

3.  Divide  9811.0047  by  .325947. 

4.  Divide  .1  by  .0001.  |       5.  Divide  10  by  .15. 

6.  Divide  6  by  .6  ;  by  .06  ;  by  .006  ;  by  .2  ;  by  .3  ;  by 
by  .003  ;  by  .5  ;  by  .05  ;  by  .005. 

Note. — 2.  When  it,  is  nece.'-sary  to  continue  the  divi.sion  farther 
than  ihe  figures  of  the  dividend  will  allow,  we  annex  ciphers,  and 
coiLsider  them  as  decimal  places  of  the  dividend. 

When  the  division  does  not  terminate,  we  annex  the  plus  sign 
to  show  that  it  may  be  continued  :  thus  .2  divided  by  .3  =  .666+. 


EXAMPLES. 


'1.  Divide  4.25  by  1.25. 

Analysis. — In  this  example  we  annex  one  0, 
and  tlien  the  decimal  places  m  the  dividend  will 
exceed  those  in  the  divisor  by  1. 


OPERATION. 

1.25)4.25(3.4 
3.75 

■~500 
500 

AnlTSA. 


2.  Divide  .2  by  .6. 

3.  Divide  37.4  by  4.5. 


4    Divide  586.4  by  375. 

5.  Divide  94.0369  by  81.032. 


Note. — 3.  When  any  decimal  number  is  to  be  divided  by  10, 
100,  1000,  &c.,  the  division  is  made  by  removing  the  decimal 
point  as  many  places  to  the  left  as  there  are  O'i-  in  the  divisor  ;  and 
if  there  be  not  .so  many  figures  on  tlie  left  of  the  decifrial  pdint 
the  deficiency  is  supplied  by  prefixing  ciphers. 

10  ]         f  2.769 

100 

1000 

10000    j 


27.69- divided  by 


\-\ 


.2769 

.02769 

.002769 


DECIMAL    KKACTIONS.  195 


1.0    ] 

^64.289 

100 

6.4289 

1000  [  =  ■{ 

.64289 

10000 

.064289 

100000 

[   .0064289 

642.89  divided  by 


QUESTIONS    IN    THE   PRECEDING    RULES. 

1  If-  I  divide  .6  dollars  among  94  men,  how  much  will 
each  leceive  ? 

2.  I  ^ave  28  dollars  to  267  persons  :  how  much  apiece  ] 

3.  Divide  6.35  by  .425. 

4.  What  is  the  quotient  of  $36.2678  divided  by  2.25  ? 

5.  Divide  a  dollar  into  12  equal  parts. 

6.  Divide  .25  of  3.26  into  .034  of  3.04  equal  parts. 

7.  How  many  times  will  .35  of  35  be  contained  in  .024 
of  24  ? 

8.  At  .75  dollars  a  bushel,  how  many  bushels  of  rye  can 
be  bought  for  141  dollars? 

9.  Bought  12  and  15  thousandths  bushels  of  potatoes  for 
33  hundredths  dollars  a  bushel,  and  paid  in  oats  at  22  hun- 
dredths of  a  dollar  a  bushel :  how  many  bushels  of  oats  did  it 
take? 

10.  Bought  53.1  yards  of  cloth  for  42  dollars  :  how  much 
was  it  a  yard  ? 

11.  Divide  125  by  .1045. 

12.  Divide  one  millionth  by  one  billionth. 

13.  A  merchant  sold  4  parcels  of  cloth,  the  first  contained 
127  and  3  thousandths  yards  ;  the  2d,  6  and  3  tenths  yards  ; 
the  3d,  4  and  one  hundredth  yards  ;  the  4th,  90  and  one 
millionth  yards  :  how  many  yards  did  he  sell  in  all '? 

14.  A  merchant  buys  three  chests  of  tea,  the  first  contains 
-60  and  one  thousandth  pounds  ;  the  second,  39  and  one  ten 

thousandth  pounds  ;  the  third,  26  and  one  tenth  pounds  :  how 
much  did  he  buy  in  all  ? 

Note. — 1.  if  there  are  more  decimal  places  in  the  divisor  than  in  the 
dividend,  what  do  you  do  !  What  will  the  figures  of  the  quotient  then 
be! 

2.  How  do  you  continue  the  division  after  you  have  brought  down  all 
the  figures  of  the  dividend  I  What  sign  do  you  place  after  the  quo- 
tient !      What  does  it  show  1 

3.  How  do  you  divide  a  decimal  fraction  by  10,  100,  1000,  «tc.  1 


VJi)  DIVISION    OF 

15.  What  is  the  sum  oi"  ^20  and  three  hundredths  ;  $4 
and  one-tenth,  $6  and  one  thousandth,  and  $18  and  one 
hundredth  1 

16.  A  puts  m  trade  $504,342  ;  B  puts  in  ;fe350.1965  ;  C 
puts  in  $100.11;  D  puts  in  $99,334;  and  E  puts  in 
$9001.32  :   what  is  the  whole  amount  put  in  1 

17.  B  has  $936,  and  A  has  $1,  3  dimes  and  1  mill  :  how 
much  more  money  has  B  than  A  1 

18.  A  merchant  buys  37.5  yards  of  cloth,  at  one  dollar 
twenty-five  cents  per  yard  :  how  much  does  the  whole 
come  to  ? 

19.  If  12  men  had  each  $339  one  dime  9  cents  and  3 
mills,  what  would  be  the  total  amount  of  their  money  ? 

20.  A  farmer  sells  to  a  merchant  13.12  cords  of  wood  at 
$4,25  per  cord,  and  13  bushels  of  wheat  at  $1,06  per  bushel  : 
he  is  to  take  in  payment  13  yards  of  broadcloth  at  $4,07  per 
yard,  and  the  remainder  in  cash  :  how  much  money  did  he 
receive  1 

21.  If  one  man  can  remove  5.91  cubic  yards  oi  earth  in  a 
day,  how  much  could  nineteen  men  remove  *? 

22.  What  is  the  cost  of  8.3  yards  of  cloth  at  $5,47  per 
yard  ? 

23.  If  a  man  earns  one  dollar  and  one  mill  per  day,  how 
much  will  he  earn  in  a  year  of  313  working  days  ? 

24.  What  will  be  the  cost  of  375  thousandths  of  a  cord  of 
wood,  at  $2  per  cord  ? 

25.  A  man  leaves  an  estate  of  $1473.194  to  be  equally 
divided  among  12  heirs  :  what  is  each  one's  portion  ? 

26.  If  flour  is  $9,25  a  barrel,  how  many  barrels  can  I  buy 
for  $1637,25  ? 

27.  Bought  26  yards  of  cloth  at  $4,37^  a  yard,  and  paid 
for  it  in  flour  at  $7,25  a  barrel  :  how  much  flour  will  pay 
for  the  cloth  1 

28.  How  much  molasses  at  22^  cents  a  gallon  must  be 
given  for  46  bushels  of  oats  at  45  cents  a  bushel  ? 

29.  Plow  many  days  work  at  $1,25  a  day  must  be  givea 
for  6  cords  of  wood,  worth  $4,12J  a  cord  ? 

30.  What  will  36.48  yards  of  cloth  cost,  if  14.25  yard 
cost  $21.3751 

31.  If  you  can  buy  13.25Z6.  of  coflee  for  $2,50,  how  much 
can  you  buy  lor  $325,50  ^ 


DECIMAL    FKACnOHB.  107 

212.    To  ciiange  a  conwion  to  a  decimal  fraction. 

The  value  of  a  fraction  is  the  quotient  of  the  numerator, 
divided  by  the  denominator  (Art  148). 

1.  Reduce  |^  to  a  decimal. 

If  we  place  a  decimal  point  after  the  5,  and  then      operation. 
write  any  number  of  O's,  after  it,  the  value  of  the         8)5.000 
numerator  will  not  be  changed  (Art.  205).  jg25 

If,  then,  we  divide  by  the  denominator,  the  quo- 
tient will  Ije  the  decimal  number  :  hence, 

Rule. — Annex  decimal  ciphers  to  the  numerator^  and 
then  divide  by  the  denominator,  pointing  off  as  in  division 
of  decimals. 

EXAMPLES. 

1,  Reduce  f|f  to  its  equivalent  decimal. 

OPERATION. 

125)635(5.08 
We  here  use  two  ciphers,  and  therefore  point  g25 

oflf  two  decimal  places  in  the  quotient. 


iOOO 
1000 


Reduce  the  following  fractions  to  decimals 


1.  Reduce  f  to  a  decimal. 

2.  Reduce  ^|  to  a  decimal. 

3.  Reduce  ^  to  a  decimal. 

4.  Reduce  i  and  yt2^' 
6.  Reduce  ^^,J|,  and  yo\o. 

6.  Reduce  ^  and  j^Wj. 

7.  Reduce  fj^-JiH- 

8.  Reduce  I.  ieiHff. 

9.  Reduce^  to  a  decimal. 

213.  A  decimal  fraction  may  be  changed  to  the  form  of  a 
vulgar  fraction  by  simply  vi^riting  its  denominatoi  (Art.  202). 


10.  Reduce  ^  to  a  decimal. 

11.  Reduce  ^^. 

12.  Reduce  ^^. 

13.  Reduce  ^^, 

14.  Reduce  yiig. 

15.  Reduce  j^, 

16.  Reduce  ^Vo- 

17.  Reduce  roW^i- 

18.  Reduce  j\%\. 


212.   How  do  you  change  a  vulgar  to  a  decimal  fraction  ? 

SIVJ.  How  do  you  change  a  decimal  to  the  form  of  a  vulgar  fraction  1 


198  DLIKUMlNAnO    DE€IMALS. 


EXAMPLES. 

1.  What  vulgar  fraction  is  equal  to  .04  ? 

2.  What  vulgar  fraction  is  equal  to  3.067  ? 

3.  What  vulgar  fraction  is  equal  to  8.275? 

4.  What  vulgar  liaction  is  equal  to  .00049  1 

DENOMINATE  DECIMALS 

214.  A  denominate  decimal  is  one  in  which  the  unit  of  the 
fraction  is  a  denominate  number.  Thus,  .5  of  a  pound,  .6  of  a 
shiUing,  .7  of  a  yard,  &c.,  are  denominate  decimals,  in  which 
the  units  are  1  pound,  1  shilling,  1  yard. 

CASE    I. 

215.  To  change  a  denominate  number  to  a  denominate 
decimal. 

I.   Change  9^.  to  the  decimal  of  a  £. 

Analysis. — The  denominate  unit  of  the  frae-  operation. 

tion  is  li:=240(/.     Then  divide  &</.   by  240:  2AQd.=£\ 

the  quotient.  .0375  of  a  pound  is  the  value  of  240)9(.0375 

9d.  in  tlie  decinjai  of  a  £  :  hence,  ^^^     £  0375 

Rule. — Reduce  the  unit  of  the  required  fraction  to  the  unit 
of  the  given  denominate  nuinher^  and  then,  divide  the  denomi- 
nate 7iU7nber  by  the  result,  and  the  quotient  will  be  the  decr'»al. 

EXAMPLES. 

1 .  Reduce  7  drams  to  the  decipial  of  a  lb.  avoirdupois. 

2.  Reduce  2Qd.  to  the  decimal  of  a  £. 

3.  Reduce  .056  poles  to  the  decimal  of  an  acre. 

4.  Reduce  14  minutes  to  the  decimal  of  a  day. 

5.  Reduce  21  pints  to  the  decimal  of  a  peck. 

6.  Reduce  3  hours  to  the  decimal  of  a  day.  * 

7.  Reduce  375678  feet  to  the  decimal  of  a  mile. 

8.  Reduce  36  yards  to  the  decimal  of  a  rod. 

9.  Reduce  .5  quarts  to  the  decimal  of  a  barrel. 

10.  Reduce  .7  of  an  ounce,  avoirdupois,  to  the  decimal  of  a 
hundred. 

214.  What  is  a  denominate  decima. 

215.  How  do  yuu  change  a  denominate  number  tu  a  deuumuiate 
dociiu'iil  1 


DENOMINATE    DECIMALS.  199 

CASE    II. 

216.  To  find  the  value  of  a  decimal  in  integets  of  a  less 
denomifiatiou . 

I.  Find  the  value  of  .890625  bushels. 

OPERATION. 

Analysis. — Multiplying  the  decimal  by  4,  (since  4  '             . 

pecks  make  a  bushel),  we  have  3.5625  pecks.    Mul- 

tiplying-  the  new  decimal  by  8,  (smce  8  quarts  make  3.562500 

a  peck),  we  have   4.5  quarts.     Then,  multiplying  8 

this  last  decimal  by  2,  (since  2  pints  make  a  quart),  4  500000 

we  have  1  pint :  lience,  *             o 

Ans.  3pk.  4qts.  Ipt.      LOOOOOO 

Rule. — I.  Multiply  the  decimal  by  that  number  which 
will  reduce  it  to  the  next  less  denomination^  pointing  off  as 
in  multiplicatix)7%  of  decimal  fractions . 

II.  Multiply  the  decimal  part  of  the  product  as  befm-e  ;  and 
so  conti?iue  to  do  until  the  decimal  is  reduced,  to  the  required 
denominations.   The  integers  at  the  left  form  the  answer 

EXAMPLES. 

1.  What  is  the  value  of  002084M.  Troyi 

2.  "What  is  the  value  of  .625  of  a  cwt»  % 

,  3  What  is  the  value  of  .625  of  a  gallon  1 

4.  What  is  the  value  of  £.3375  l 

5.  What  is  the  value  of  .3375  of  a  ton  ? 

6.  What  is  the  value  of  .05  of  an  acre? 

7.  What  is  the  value  ol..675  pipes  of  wine  1 

6.  What  is  the  value  of  .125  hogsheads  of  beer  1 

9.  What  is  the  value  of  .375  ol  a  year  of  365  days  ] 

10.  What  is  the  value  of  .085  of  a  X  ? 

1 1.  What  is  the  value  of  .86  of  a  cwt.  1 

12.  From  .62  of  a  day  take  .32  of  an  hour. 

13.  What  is  the  value  of  1.089  miles'? 

14.  What  is  the  value  of  .09375  of  a  pound,  avoirdupois  1 

15.  What  is  the  value  of  .28493  of  a  year  of  365  days  ] 

16.  What  is  the  value  of  £1.046  ? 

17.  What  is  the  value  of  £1.88  ? 

216.  How  do  you  find  the  value  of  a  decimal  in  integers  of  a  iese 
dciiominatioa  \ 


200*  DKNOMINATIC    DliCIMALB. 


CASE  in. 

217.  To  reduce  a  cpmpmmd  denominate  number  to  a 
decimal  or  mixed  number. 

1.  Reduce  £1  4s.  9J<i.  to  the  decimal  of  a£. 

Analysis. — Reducing  the  |f/.  to  a  decimal  operation 

(Art.  21r»).  and  annexing  the  result  to  the  9rf.,  3.7  _  75/* 

we  have  9  15d.     Dividing  9.75c/.  by  12,  (since  fA  i'~  q  757 
12  pence  =l.v.),  and  annexing  tlie  quotient  to  ^"^  " 

the  45.  we  have  4.81255.  Then,  dividing  by  20  12^9.756^ 

(since   2i'5  — £1,)  and  annexing  the  quotient  20U  812''><j 
to  the  £) ,  we  have  £1 .240625 :  ^ 

Ans.  £1  45.  9|6Z.  =  1.240625£. 

Rule  —  Divide  the  loivest  denomination  by  as  many  units 
as  makt  a  unit  of  the  next  higher,  and  annex  the  quotient 
as  a  decimal  to  that  higher  :  then  divide  as  before,  and  so 
continui  to  do,  until  the  decimal  is  reduced  to  the  required 
denomination. 

EXAMPLES. 

1.  Reduce  'iwk.  &da.  5hr.  30m.  455.  to  the  denomination 
of  a  week. 

2.  Reduce  2/6.  5oz,  12/?r6'i.  \6gr.,  to  the  denomination  of  a 
pound. 

3.  Reduce  3  feet  9  inches  to  the  denomination  of  yards. 

4.  Reduce  lib.  i2dr.,  avoirdupois,  to  the  denomination  of 
pounds. 

5.  Reduce  5  leagues  2  furloj;igs  to  the  denomination  of 
leagues. 

6.  Reduce  4:bu.  Spk.  "iqt.  Ipt.  to  the  denomination  of 
bushels. 

7.  Reduce  5oz.  ISpwt.  I2gr.  to  the  decimal  of  a  pound. 

8.  Reduce  15cwt.  2>qr.  2^lb.  to  the  decimal  of  a  ton. 

9.  Reduce  5 A.  SR.  2lsq.  rd.  to  the  denomination  of  acres. 

10.  Reduce  11  pounds  to  the  decimal  of  a  ton. 

11.  Reduce  3da.  12^5^6*.  to  the  decimal  of  a  week. 

12.  Reduce  lAbu.  ofqt.  to  the  decimal  of  a  chaldron, 

13.  Reduce  77n.  Ifur.  \r.  to  the  denomination  of  miles. 


217.   How    do   you    reduce   a   compound  denomiaate  nuuiber  to 
decimal  ' 


ANALYSIS.  201 


ANALYSIS. 

218.  An  analysis  of  a  proposition  is  an  examination  of  its 
separate  parts,  and  their  connections  with  each  other. 

The  sohjtion  of  a  question,  by  analysis,  consists  in  an  exami- 
nation of  its  elements  and  of  the  relations  which  exist  between 
these  elements.  "We  determine  the  elements  and  the  rela- 
tions which  exist  between  them,  in  each  case,  by  examining 
the  nature  of  the  question. 

In  analyzing,  we  reason  from  a  given  uumber  to  its  unity 
and  then  from  this  unit  to  the  required  number. 

EXAMPLES. 

1.  If  9  bushels  of  wheat  cost  18  dollars,  what  will  27 
bushels  cost  ? 

Analysis. — One  bushel  of  wheat  will  cost  one  ninth  as  much  as 
9  bushels.  Since  9  bushels  cost  18.  dollars,  1  bushel  will  cost  ^ 
of  18  dollars,  or  2  dollars  ;  27  bushels  will  cost  27  times  as  much 
as   1    bushel:  that   is.  27  times  \  of  18  dollars,   or  54  dollars: 


cost  54  dollars. 

OPERATION. 


M^      1      27      ^^, 
-f  X;rX  — .=  $54;     Or, 
19       1 


IS  3 


54  Ans, 


Note. — 1.  We  indicate  the  operations  to  be  performed,  and 
then  cancel  the  equal  factors  (Art.  141). 

219.  Although  the  currency  of  the  United  States  is  ex- 
pressed in  dollars  cents  and  mills,  still  in  most  of  the  States 
the  dollar  (always  valued  at  100  cents),  is  reckoned  in  shil- 
lings and  pence  ;  thus, 

In  the  New  England  States,  in  Indiana,  Illinois,  Missouri,  Vir- 
ginia, Kentucky,  Tennessee,  Mississippi  and  Texas,  the  dollar  is 
reckoned  at  6  shillings  :  In  New  York,  Ohio  and  Michigan,  at  8 
shillings  :  In  New  Jersey,  Pennsylvania,  Delaware  and  Mary- 
land, at  Is.  6d :  In  South  Carolina,  and  Georgia,  at  45.  8d.  :  In 
Canada  and  Nova  Scotia,  at  5  shillings. 

218.  What  is  an  analysis  ]  In  what  does  the  solution  of  a  question 
by  analysis  consist  ?  How  do  we  determine  the  elements  and  theii 
relations  !     How  do  we  rea&ou  in  analyzing  1 


202 


ANALYSTS. 


Note. — In  many  of  the  States  the  retail  price  of  articles  is  given 
in  shillings  and  pence,  and  the  result,  or  cost,  required  in  doilarB 
and  cents. 

2.  What  will  12  yards  of  cloth  cost,  at  5  shillings  a  yard, 
New  York  currency  ? 

Analysis. — Since  1  yard  cost  5  shillings  12  yards  will  cost  12 
times  5  shillings,  or  60  shillings  :  and  as  8  shillings  make  1  dollar. 
New  York  cuiiency,  there  will  be  as  many  dollars  as  8  is  contain 
ed  times  in  60=$7i. 


OPERATION. 


6X12-H8  =  $7,50;       Or, 


5 


15=:i«:^|;7,50. 


$7,50. 


Note. — The  fractional  part  of  a  dollar  may  always  be  reduced 
to  cents  and  mills  by  annexing  two  or  three  ciphers  to  the  nume- 
rator and  dividing  by  the  denominator  ;  or,  which  is  more  conve- 
nient in  practice,  annex  the  ciphers  to  the  dividend  and  continue 
the  division. 

3.  What  will  be  the  cost  of  66  bushels  of  oats  at  3&'.  3d.  a 
bushel,  New  York  currency  ? 


4 

$ 

00  ^ 
13 

OPERATION. 

4 

4 

91 

Or,                   4_ 

91 

22,75. 

^22,75  Ans. 

Note. — When  the  pence  is  an  aliquot  part  of  a  shilling-  the 
price  may  be  reduced  to  an  improper  fraction,  which  will  be  the 
multiplier:  thus.  35.  3d.  =  3^s.=  ^s.  Or:  the  shillings  and  ])ence 
may  be  reduced  to  pence  :  thus,  'Ss.  3d.  =  39d.,  in  which  case  the 
product  will  be  pence,  and  must  be  divided  by  96,  the  number  of 
pence  in  1  dollar  :  hence, 

220.   To  find  the  cost  of  articles  in  dollars  and  cents. 


219.  In  what  is  the  currency  of  the  States  expressed  1 
the  currency  of  the  States  often  reckoned  ! 

230.  How  do  you  find  the  cost  of  a  commoditv 


In  what  if 


ANALYSIS. 


Wii 


Multiply  Ihe  commodity  by  the  price  and  divide  the  p-oduct 
by  the  value  of  a  dollar  reduced  io  the  sam.e  denominational 
unit. 

4.  What  will   18  yards  of  satinet  cost  at  3s.   9c?.    a  yard 
Pennsylvania  currency  1 


u 


OPERA! 

'ION. 

u' 

Q 

1$ 

S0 

n 

t 

Or, 

4$ 

^9. 

$9  Ans 

Note. — The  ahove  rule  will  apply  to  the  currency  in  any  of 
ihe  States.  In  the  last  example  the  multiplier  is  "^s.  9(i.  =  3i5 
=  1^.9.  or  45rf.     The  divisor  is  75.  Qd.  =  1U.=  ^s.^dOd. 

5.  What  will  7^/6.  of  tea  cost  at  65.  Sd.  a  pound,  Ncm 
Eng'und  currency  ? 


OPERATION. 


it  U ' 


Or, 


$0 


10 


25 


3  !   25  =  25  r=  $8,333-1- 


5.333-|-^w«. 


6.  What  will  be  the  cost  of  '[20yds.  of  cotton  cloth  at   1*. 
6d.  a  yard,  Georgia  currency  ] 

7.  What  will  be  the  cost  in  New  York  currency? 

8.  What  will  be  the  cost  in  New  England  currency  ? 

9.  What  will  be  the  cost  of  15   bushels  of  potatoes  at  3s 
6fl?.,  New  York  currency  ? 

10.  What  will  it  cost  to  build  148   feet  of  w^all  at  Is.  8d.. 
per  loot,  N.  Y.  currency  ? 

11.  What  will  a  load  of  M'heat,  containing  46 J  bushels, 
come  to  at  lO.s.  8d.  a  bushel,  N.  Y.  currency  ] 

12.  What  will  7  yards  of  Irish  linen  cost  at  3s.  4c/.  a  yard, 
Penn.  currency  ? 

13.  How  many  pounds  of  butter  at  Is.  4.d.  a   pound  must 
be  given  for  12  gallons  of  molasses  at  2.v.  8d.  a  gallon  ? 


204  AWALYISIS. 


OPERATION. 

I   12  ,   12  - 

$ \  $         '  H\n       ^ 

^  I     ^  0^'  \2Ub, 

I  24/6. 
Note. — The  same  rule  applies  in  the  last  example  as  in  the 
preceding  ones,  except  that  the  divisor  is  the  price  of  tlie  article 
received  in  payment,  reduced  to  the  same  unit  as  the  price  of  the 
article  bought. 

14.  What  will  be  the  cost  of  \2cwt.  of  sugar  at  \^d.  per  lb. 
N.  Y.  currency  ? 

Note. —  Reduce  the  cwts.  to  Ihs.  by 
multiplying  by  4  and  then  by  25.  Then 
multiply  by  the  price -per  pound,  and 
then  divide  by  the  value  of  a  dollar  in 
the  required  currency,  reduced  to  the 
same  denomination  as  the  price. 

$112,50 

15.  What  will  be  the  cost  of  9  hogsheads  of  molasses  at  Is. 
3c?.  per  quart,  N.  E.  currency  '\ 

16.  How  many  days  work  at  75.  6r/.  a  day  must  be  given 
ibr  12  bushels  of  apples  at  3s. 9^.  a  bushel? 

17.  Farmer  A  exchanged  35  bushels  of  barle} ,  worth  65. 
4:d.,  with  farmer  B  for  rye  worth  7  shillings  a  bushel  :  how 
many  bushels  of  rye  did  farmer  A  receive  ? 

18.  Bought  the  following  bill  of  goods  of  Mr.  Merchant: 
what  did  the  whole  amount  to,  JST.  Y.  currency  1 

12^  yards  of  cambric  at  Is.  ^d.  per  yard. 

8       "            ribbon  "  2s.  &d. 

21       "            calico  •'   Is.  M. 

6       *'            alpaca  "  5s.  dd.         " 

4    gallons     molasses  **  3s.  5d.  per  gallon. 

2^  pounds     tea  "  6s.  6d.  per  pound. 

30    .  "           sugar  "  9d.     "         " 

19.  If  f  of  a  yard  of  cloth  cost  $3,20,  what  will  ||  of  a 
yard  cost  1 

Analysis. — Since  5  eighths  of  a  yard  of  cloth  costs  $3,20. 1  eighth 
of  a  yard  will  cost  ^  of  $3,20 ;  and  1  yard,  or  8  eighths,  will  cost 
8  times  as  much,  or  |  of  $3,20  ;  ^|  of  a  yard  will  cost  \^  as  much 
as  1  yard,  or  {^  of  |  of  $3,20 =$4.80. 


160      J 


ANALYSIS.  '"205 

OPERATION. 

3  ftii'irtl'GO 


$$;^0X^XyX~  =  S4.8O.     Or,  0 


^ 


10 


10 


$4,80. 


20.  If  3j  pounds  of  tea  cost  3^  dollars,  what  will  9  poundg 
cost  1 

Note. — Reduce  the  mixed  numbers  to  improper  fractions,  and 
then  apply  the  same  mode  of  reatsoning  as  in  tlie  preceding  ex- 
ample. 

21.  What  will  8i  cords  of  wood  cost,  if  2|-  cords  cost  7-1 
dollars  ? 

22.  If  6  men  can  build  a  boat  in  120  days,  how  long  will 
it  take  24  men  to  build  it  ? 

Analysis. — Since  6  men  can  build  a  boat  in  120  days,  it  will 
take  1  man  6  times  120  days,  or  720  days,  and  24  men  can  build 
it  in  ^  of  the  time  that  1  man  will  require  to  build  it,  or  -jlj-  of  6 
times  120,  which  is  30. 

OPERATION. 


120x6-^24=30  days.     Or,       . 


0 


Ans.     30  da^s. 

23.  If  7  men  can  dig  a  ditch  in,  21  days,  how  many  men 
will  be  required  to  dig  it  in  3  days  ? 

24.  In  what  time  will  12  horses  consume  a  bin  of  oats, 
that  will  last  21  horses  6|-  weeks  ? 

25.  A  merchant  bought  a  number  of  bales  of  velvet,  each 
containing  129i|^  yards,  at  the  rate  of  7  dollars  for  5  yards, 
and  sold  them  at  the  rate  of  1 1  dollars  for  7  yards  ;  and 
gained  200  dollars  by  the  bargain  :  how  many  bales  were 
there  ? 

Analysis. — Since  he  paid  7  dollars  for  5  yards,  for  1  yard  he 
paid  ^  of  $7  or  I  of  1  dollar  ;  and  since  he  received  11  dollars  for 
7  yards,  for  1  yard  he  received  \  oi  11  dollars  or  ^  of  1  dollar. 
He  gained  on  1  yard  the  difference  between  ^  and  ^=-^  of  a  dol- 
lar. Since  his  whole  gain  was  200  dollars,  he  had  as  many  yards 
as  the  gain  on  one  yard  is  contaiiied  times  in  his  whole  gain,  or 
as  ^  is  contained  times  in  200.  And  there  were  as  many  bales 
as  129^.  (the  number  of  yards  in  one  bale),  is  contained  times  in 
the  whole  number  of  yards  ^<^  ;  which  gives  i)  bale*. 


200 


ANALYSIS. 


$000 


*^00 


OPERATION. 

129^\—^^,  number  ofyardsin  a  ba.e  :  ^  ^^^^^^  ^ 

2004-3^5  =—^,  "^'hol*^  number  of  yards  :  i00 

Ui^o o_^3|p_o  ^  9  bales.  ^^..     |  9  bales, 

2G.  Suppose  a  number  of  bales  of  cloth,  each  contairiiuy 
133^  yards,  to  be  bought  at  the  rate  of  12  yards  for  1 1  doJ 
lars,  and  sold  at  the  rate  of  8  yards  for  7  dollars,  and  the 
loss  in  trade  to  be  ^100  :  how  many  bales  are  there? 

27.  If  a  piece  of  cloth  9  feet  long  and  3  feet  wide,  contain 
3  square  yards  ;  how  long  must  be  a  piece  of  cloth  ihat  is  2| 
feet  wide  be,  to  contain  the  same  number  of  yards? 

28.  A  can  mow  an  acre  of  grass  in  4  hours,  B  in  6  hours, 
and  C  in  8  hours.  How  many  days,  working  9  hours  a  day, 
would  they  require  to  mow  39  acres  ? 

Analysis. — Since  A  can  mow  an  acre  in  4  hours,  B  in  6  hours, 
and  C  in  8  hours,  A  can  mow  \  of  an  acre,  B  ^  of  an  acre,  and 
C  ^  of  an  acre  in  1  hour,  Together  they  can  mow  i+i+i=M 
of  an  acre  in  1  hour.  And  since  they  can  mow  13  twenty-fourths 
of  an  acre  in  1  hour,  they  can  mow  1  twenty-fourth  of  an  acre 
in  ^  of  1  hour;  and  1  acre,  or  f|,  in  24  times  ^==f  |  of  1  hour: 
and  to  mow  39  acres,  they  will  require  39  times  f  |=r  \2/  hours, 
which  reduced  to  days  of  9  hours  each,  gives  8  days. 


OPERATION. 


8       $ 

U      $0       1 


n 


—  X  T  =^  ^  days.  Or 
1       0  ^ 


Ans. 


u 


n 


8  days. 


29.  A  can  do  a  piece  of  work  in  4  days,  and  B  can  do  the 
same  in  6  days  ;  in  what  time  can  they  both  do  the  work  if 
they  labor  together  ? 

30.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  it  \ 

Analysis. — If  6  men  can  do  a  piece  of  work  in  10  day.'^,  1  man 
will  require  6  times  as  long,  or  60  days  to  do  tiie  t^ame  work. 
Five  men  will  require  but  onc-fifLh  as  long  as  one  man.  or  bO-f-'O 
=  12  days. 


10x6-^5=12  days. 


ANALYSIS. 

OPERATION. 

u 

$ 

6 

Ans. 

12  days 

207 


31.  Three  men  together  can  perform  a  piece  of  work  in  9 
days.  A  alone  can  do  it  in  18  days,  B  in  27  days  ;  in  what 
time  can  C  do  it  alone  1 

32.  A  and  B  can  build  a  wall  on  one  side  of  a  square 
piece  of  ground  in  3  days  ;  A  and  C  in  4  days  ;  B  and  C  in 
G  days  :  what  time  will  they  require,  working  together,  to 
complete  the  wall  enclosing  the  square  I 

33.  Three  men  hire  a  pasture,  lor  M'hich  they  pay  66  dol- 
lars. The  first  puts  in  2  horses  3  weeks  ;  the  second  6  horses 
for  2^  weeks  ;  the  third  9  horses  for  1^  weeks  :  how  much 
ought  each  to  pay  1 

Analysis. — The  pasturage  of  2  horses  for  3  weeks,  would  be  the 
same  as  the  pasturage  of  1  horse  2  times  3  weeks,  or  6  weeks: 
that  of  six  horf^es  2^  weeks,  the  same  as  for  1  horse  6  times  2-| 
weeks,  or  15  weeks  ;  and  that  of  9  horses  \^  weeks,  the  same  as 
1  horse  tor  9  times  1-|^  weeks,  or  12  weeks.  The  three  persons  had 
an  equivalent  for  the  pasturage  of  1  horse  for  6  +  1 5  + 1 2  =  33  weeks  ] 
therefore,  the  first  must  pay  ^,  the  second  ^,  and  the  third 
\^  of  66  dollars. 

OPERATION. 

3    x2i=:6;         then       $66x^=$12.     1st. 
2^X6  =  15;  "  $66xifrz:$30.     2d. 

11x9  =  12;  "  $66x^1=:  $24.     3d. 

34.  Two  persons,  A  and  B.  enter  into  partnership,  and  gain 
$175.  A  puts  in  75  dollars  for  4  months,  and  B  puts  in  100 
dollars  for  6  months  :  what  is  each  one's  share  of  the  gain  ? 

35.  Three  men  engage  to  build  a  house  for  580  dollars. 
The  first  one  employed  4  hands,  the  second  5  hands,  and  the 
third  7  hands.  The  first  man's  hands  worked  three  times  as 
many  days  as  the  third,  and  the  second  man's  hands  tM'ice  as 
many  days  as  the  third  man's  hands  :  how  much  must  each 
receive  1 


208 


.ANALYSIS. 


36.  If  8  students  spend  $192  in  6  months,  how  much  will 
12  students  spend  in  20  months  ? 

Analysis. — Since  8  students  spend  $192,  one  student  will  spend 
J  of  $192,  in  6  months;  in  1  month  1  student  will  spend  ^  of  \ 
of  $192= $4.  Twelve  students  will  spend,  in  1  month,  12  times 
as  much  as  1  student,  and  in  20  months  they  will  spend  20  times 
as  much  as  in  1  month. 


OPERATION. 


24 


1     1     I^     20     ^^^^ 

1  ^6^^^T^T=^^^^ 


m 


20 


48 


$960.  Ans, 


37.  If  6  men  can  build  a  waJl  80  feet  long,  6  feet  wide, 
and  4  feet  high,  in  15  days,  in  what  time  can  18  men  build 
one  240  feet  long,  8  feet  wide,  and  6  leet  high  ? 

Analysis. — Since  it  takes  6  men  15  days  to  build  a  wall,  it 
will  take  1  man  6  times  io  days,  or  90  days,  to  build  the  same 
wall.  To  build  a  wall  1  foot  long,  will  require  ^  as  long  as  to 
build  one  80  feet  long;  to  build  one  1  foot  wide,  \  as  long  as  to 
build  one  4  feet  wide ;  and  to  build  one  1  foot  high,  \  as  long  as 
to  build  one  6  feet  high.  18  men  can  build  the  same  wall  in  ^ 
of  the  time  that  one  man  can  build  it:  but  to  build  one  240  feet 
long,  will  take  them  240  times  as  long  as  to  build  one  1  foot  in 
length;  to  build  one  8  feet  wide,  8  times  as  long  as  to  build  one 
1  foot  wide,  and  to  build  one  6  feet  high,  6  times  as  long  as  to 
build  one  1  foot  high. 


OPERATION. 

%     2 
15x0       1       1      1       1      ^^0      0     $ 

'- X  "T-r  X  —;  X  T  X  ~r  X     z      X~X~  —  oU. 

1  $0       ^       $       1$  I  I       1 


$ 


0 


15 


$  2 


Ans.  I  30  days. 

38.  If  96Z^5.  of  bread  be  sufficient  to  serve  5  men  12  days, 
how  many  days  will  6716.  serve  19  men? 


ANALYSIS. 

39.  If  a  man  travef  220  miles 
hours  a  day,  in  how  many  days  wil 
travelling  16  hours  a  day  i 

40.  If  a  family  of  12  persons  consume  a  certain  quantity 
of  provisions  in  6  days,  how  long  will  the  same  provisions 
last  a  I'amily  of  8  persons  ? 

41.  If  9  men  pay  $135  for  5  weeks'  board,  how  much 
must  8  men  pay  for  4  weeks'  board  I 

42.  If  10  bushels  of  wheat  are  equal  to  40  bushels  of 
corn,  and  28  bushels  of  corn  to  56  pounds  of  butter,  and  39 
pounds  of  butter  to  1  cord  of  wood  ;  how  much  wheat  is  12 
cords  of  wood  worth  ? 

Analysis. — Since  10  bushels  of  wheat  are  worth  40  bushels  of 
corn,  1  bushel  of  corn  is  worth  ^  of  10  bushels  of  wheat,  or 
\  of  a  bushel ;  28  bushels  are  worth  28  limes  ^  of  a  bushel  of 
wheat,  or  7  bushels :  since  28  bushels  of  corn,  or  7  bushels  of 
wheat  are  worth  56  pounds  of  butter.  1  pound  of  butter  is  worth 
^  of  7=^  of  a  bushel  of  wheat,  and  39  pounds  are  worth  30 
times  as  much  as  1  pound,  or  39x-^=^  bushels  of  wheat;  and 
since  39  pounds  of  butter,  or  ^  bushels  of  wheat  are  worth  1  cord 
of  wood,  12  cords  are  worth  12  times  as  much,  or  12X^  =  ^8"^ 
Dushels. 

OPEKATION. 


3  ^    .. 

4  2 


10 

n 

39 

n 


Note. — Always  commence  analyzing  from  the  term  which  is 
of  the  same  name  or  kind  as  the  required  answer. 

43.  If  35  women  can  do  as  much  work  as  20  boys,  and 
16  boys  can  do  as  much  as  7  men  :  how  many  women  can 
do  the  work  of  1 8  men  ? 

44.  If  36  shillings  in  New  York  are  equal  to  27  shillings 
in  Massachusetts,  and  24  shillings  in  Massachusetts  are  equal 
tx)  30  shilHngs  in  Pennsylvania,  and  45  shillings  in  Pennsyl- 
vania are  equal  to  28  shillings  in  Georgia  ;  how  many  shil- 
lini^s  in  Georgia  are  equal  to  72  shillings  in  New  York  ? 

14 


210  PROMISCUOUS   EXAMPLES 

PROMISCUOUS    EXAMPLES    IN    ANALYSIS. 

1.  How  many  sheep  at  4  dollars  a  head  must  I  give  fcr  6 
cows,  worth  1 2  dollars  apiece  ? 

2.  If  7  yards  of  cloth  cost  ^^49,  what  will  16  yards  cost  ? 

3.  If  36  men  cun  build  a  house  in  16  days,  how  long  will 
it  take  12  men  to  build  it  ? 

4.  If  3  pounds  of  butter  cost  71  shillings,  what  will  12 
pounds  cost  ? 

0.  If  5i  bushels  of  potatoes  cost  S2|,  how  much  will  1 2^ 
bushels  cost  ? 

6.  How  many  barrels  of  apples,  worth  12  shillings  a  barrel, 
will  pay  for  16  yards  of  cloth,  worth  9s.  6d.  a  yard  ? 

7.  If  31i  gallons  of  molasses  are  worth  $9f,  what  are  5 J 
gallons  worth"? 

8.  What  is  the  value  of  24j  bushels  of  corn,  at  5s.  Id.  a 
bushel,  New  York  currency  ? 

9.  How  much  rye,  at  8s.  3d.  per  bushel,  must  be  given 
for  40  gallons  of  whisky,  worth  2s.  dd.  a  gallon? 

10.  If  it* take  44  yards  of  carpeting,  that  is  li  yards  wide, 
to  cover  a  floor,  how  many  yards  of  J  yards  wide,  will  it 
take  to  cover  the  same  floor] 

11.  Li'  a  piece  of  wall  paper,  14  yards  long  and  1^  feet 
wide,  will  cover  a  certain  piece  of  wall,  how  long  must  an- 
other piece  be,  that  is  2  feet  wide,  to  cover  the  same  wall  ? 

12.  If  5  men  spend  $200  in  160  days,  how  long  will  $300 
last  12  men  at  the  same  rate  ? 

13.  If  1  acre  of  land  cost  J  of  f  of  f  of  $50,  what  will  3  J 
acres  cost  ? 

14.  Three  carpenters  can  finish  a  house  in  2  months  ;  two 
of  them  can  do  it  in  2i  months  :  how  long  will  it  take  tho 
third  to  do  it  alone  1 

15.  Three  persons  bought  2  barrels  of  flour  lor  15  dollars 
The  first  one  ate  from  them  2  months,  the  second  3  months 
and  the  third  7  months  :  how  much  should  each  pay  ? 

16.  What  quantity  of  beer  will  serve  4  persons  18-J  days 
if  6  persons  drink  7^  gallons  in  4  days  ? 


IN    ANALYSIS.  211 

17.  If  9  persons  use  If  pounds  of  tea  in  a  month,  how 
much  will  10  persons  use  in  a  year  ? 

18.  If  ^  of  f  of  a  gallon  of  wine  cost  f  of  a  dollar,  what 
will  5J  gallons  cost  ? 

19.  How  many  yards  of  carpeting,  If  yards  wide,  will  it 
take  to  cover  a  floor  that  is  4f  yards  wide  and  6  and  three- 
fifths  yards  long  ? 

20.  Three  persons  bought  a  hogshead  of  sugar  containing 
413  pounds.  The  first  paid  $2^  as  often  as  the  second  paid 
831,  and  as  often  as  the  third  paid  $4  :  what  was  each  one's 
share  of  the  sugar  ? 

21.  A,  wath  the  assistance  of  B,  can  build  a  wall  2  feet 
wide,  3  feet  high,  and  30  feet  long,  in  4  days  ;  but  with  the 
assistance  of  C,  they  can  do  it  in  3^  days  :  in  how  many  days 
can  C  do  it  alone  1 

22.  If  two  persons  engage  in  a  business,  where  one  advances 
$875,  and  the  other  $625,  and  they  gain  $300,  what  is  each 
one's  share  1 

23.  A  person  purchased  ^  of  a  vessel,  and  divided  it  into  5 
equal  shares,  and  sold  each  of  those  shares  for  $1200  :  what 
was  the  value  of  the  whole  vessel  t 

24.  How  many  yards  of  paper,  j  of  a  yard  wide,  will  be 
sufficient  to  paper  a  room  1 0  yards  square  and  3  yards  high  1 

25.  What  M'ill  be  the  cost  of  4.6lbii.  of  coffee,  New  Jersey 
currency,  if  9lbs.  cost  27  shillings  ? 

26.  Wliat  will  be  the  cost  of  3  barrels  of  sugar,  each  weigh- 
ing 2cwl.  at  lOd.  per  pound,  Illinois  currency? 

27.  If  12  men  reap  80  acres  in  6  days,  in  how  many  days 
will  25  men  reap  200  acres  ? 

28.  If  4  men  are  paid  24  dollars  for  3  days'  labor,  how 
many  men  may  be  employed  16  days  for  $96  ? 

29.  If  $25  wall  supply  a  family  with  flour  at  $7,50  a  bar- 
rel for  2|-  months,  how  long  would  $45  last  the  same  family 
when  flour  is  worth  $6,75  per  barrel  1 

30.  A  wall  to  be  built  to  the  height  of  27  feet,  was  raised 
to  the  height  of  9  feet  by  12  men  in  6  days  :  how  many  men 
must  be  employed  to  finish  tlie  wall  in  4  days  at  the  same 
rute  of  work  in''  l 


212  PKOMISCUOU^    EXAMPLES. 

31.  A,  B  and  C,  sent  a  drove  of  hogs  to  market,  of  which 
A  owned  lOo,  B  75,  and  C  120.  On  the  way  60  died  : 
how  many  must  each  lose '? 

32.  Three  men,  A,  B  and  C,  agree  to  do  a  piece  of  work, 
for  which  they  are  to  receive  $315,  A  works  8  days,  10^ 
hours  a  day  ;  B  9-J  days,  8  hours  a  day  ;  and  C,  4  days,  12 
hours  a  day  :   what  is  each  one's  share  ? 

33.  If  10  barrels  of  apples  will  pay  for  5  cords  of  wood, 
and  12  cords  of  wood  for  4  tons  of  hay,  how  many  barrels  of 
apples  will  pay  for  9  tons  of  hay  ? 

34.  Out  of  a  cistern  that  is  |-  full  is  drawn  140  gallons, 
"when  it  is  found  to  be  |  full  :   how  much  does  it  hold  ? 

35.  If  .7  of  a  gallon  of  wine  cost  $2,25,  what  will  .25  of  a 
gallon  cost? 

36.  If  it  take  5.1  yards  of  cloth,  1.25  yards  wide,  to  make  a 
gentleman's  cloak,  how  much  surge,  ^  yards  wide,  will  be 
required  to  hue  it  ? 

37  A  and  B  have  the  same  income.  A  saves  i  of  his 
annually  ;  but  B,  by  spending  '1>200  a  year  more  than  A,  at 
the  end  of  5  years  finds  himself  $160  in  debt :  what  is  their 
income  ? 

38.  A  father  gave  his  yomiger  son  $420,  which  was  J  of 
what  he  gave  to  his  elder  son  ;  and  3  times  the  elder  son's 
portion  was  ^  the  value  of  the  father's  estate  :  what  was  the 
value  of  the  estate  ? 

39.  Divide  $176,40  among  3  persons,  so  that  the  first  shall 
have  twic^  as  much  as  the  second,  and  the  third  three  times 
as  much  as  the  first  :  w^hat  is  each  one's  share  1 

40.  A  gentleman  having  a  purse  of  money,  gave  ^  of  it  foi 
a  span  of  horses  ;  j  of  |^  of  the  remainder  for  a  carriage  ; 
when  he  found  that  he  had  but  $100  left :  how  much  was  in 
his  purse  before  any  was  taken  out  ? 

41.  A  merchant  tailor  bought  a  number  of  pieces  of  cloth, 
each  containing  25^^  yards,  at  the  rate  of  3  yards  for  4  dol- 
lars, and  sold  them  at  the  rate  of  5  yards  for  13  dollars,  and 
gained  by  the  operation  96  dollars  :  how  many  pieces  did  he 
buy  i 


.EATIO   AND   PKOPORTION.  213 


RATIO    AND    PROPORTION. 

221.  Two  numbers  having  the  same  unit,  may  be  com- 
pared in  two  ways : 

Ist.  By  considering  how  much  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference  ;  and, 

2d.  By  considering  how  many  times  one  is  contained  in  the 
other,  which  is  shown  by  their  quotient. 

In  comparing  two  numbers,  one  with  the  other,  by  means 
of  their  difference,  the  less  is  always  taken  from  the  greater. 

In  comparing  two  numbers,  one  with  the  other,  by  means 
of  their  quotient,  one  of  them  must  be  regarded  as  a  standard 
which  measures  the  other,  and  the  quotient  which  arises  by 
dividing  by  the  standard,  is  called  the  ratio. 

222.  Every  ratio  is  derived  from  two  terms:  the  first  is 
called  the  antecede?it,  and  the  second  the  consequent ;  and  the 
two,  taken  together,  are  called  a  couplet.  The  antecedent  will 
be  regarded  as  the  standard. 

If  the  numbers  3  and  12  be  compared  by  then-  difierence, 
the  result  of  the  comparison  yill  be  9  ;  for,  12  exceeds  3  by  9. 
If  they  are  compared  by  means  of  their  quotient,  the  result 
will  be  4;  for,  3  is  contained  in  12,  4  times:  that  is, 
3  measuring  12,  gives  4. 

223.  The  ratio  of  one  number  to  another  is  expressed  in 
two  ways  : 

\st.  By  a  colon  ;  thus,  3  :  12  ;  and  is  read,  3  is  to  12  ;  or, 
3  measuring  12. 

i2 
2d.  In  a  fractional  form,  as  —  ;  or,  3  measuring  12. 

221.  In  how  many  ways  may  two  numbers,  having  the  same  unit,  &e 
compared  with  each  other  1  If  you  compare  by  their  difference,  how  do 
you  find  it  ^  If  you  compare  by  the  quotient,  how  do  you  regard  one  of 
the  numbers  "^     What  is  the  ratio  1 

222.  From  how  many  terms  is  a  ratio  derived  ?  WTiat  is  the  first 
tf  rm  called  ?     "Wliat  is  tlie  second  called  1     Which  is  the  standard  1 

*'.43    How  may  the  ratio  of  two  iituiibcrs  be  expresi^ed  ^     Hinv  rcuJ  ' 


214  RATIO    AND    PBOPOETION. 

224.  If  two  couplets  have  the  same  ratio,  their  tenns  are 
Bdid  to  be  proportional  :  the  couplets 

3     :     12     and     1     :     4 

have  the  same  ratio  4 ;  hence,  the  terms   are  proportional, 
and  are  written, 

3     :     12     :     :     1     :     4 

by  simply  placing  a  double  colon  between  the  couplets.     The 
terms  are  read 

3  is  to  12       as        1  is  to  4, 
and  taken  together,  they  are  called  a  lyroportion  :  hence, 

A  proportion  is  a  comparison    of  the   terms  of  two  equal 
ratios.* 


224.  If  two  couplets  have  the  same  ratio,  what  is  said  of  the  tenns  .' 
Hov/  are  they  written  1     How  read  1     What  is  a  proportion  ? 

*  Some  authors,  of  high  authority,  make  the  consequent  the  stand- 
ard and  divide  the  antecedent  by  it  to  determine  the  ratio  of  the  couplet. 

The  ratio  3  :  12  is  the  same  as  that  of  1:4  by  both  methods  ; 
for,  if  the  antecedent  be  made  the  standard,  the  ratio  is  4  ;  if  the  conse- 
quent be  made  the  standard,  the  ratio  is  one;fourth.  The  question  is, 
which  method  should  be  adopted  I 

The  unit  1  is  the  number  from  w^iich  all  other  numbers  are  derived, 
and  by  which  they  are  measured. 

The  question  is,  how  do  we  most  readily  apprehend  and  express  the 
relation  between  1  and  4  I  Ask  a  child,  and  he  will  answer,  "  the  dif- 
ference is  3  "  But  when  you  ask  him,  "  how  many  I's  are  there  in 
4  '"  he  will  answer,  "  4,"  using  1  as  the  standard. 

Thus,  we  begin  to  teach  by  using  the  standard  1  :  that  is,  by  dividing 
4  by  1. 

Now,  the  relation  between  3  and  12  is  the  same  as  that  between  1 
and  4  ;  if  then,  we  divide  4  by  1,  we  must  also  divide  12  by  3.  Do  wc, 
indeed,  clearly  apprehend  the  ratio  of  3  to  12,  until  we  have  referred  to 
1  as  a  standard  ]  Is  the  mind  satisfied  until  it  has  clearly  perceived  that 
the  ratio  of  3  to  12  is  the  same  as  that  of  1  to  4  1 

In  the  Rule  of  Three  we  always  look  for  the  result  in  the  4th  term. 
Now,  if  we  wish  to  find  the  ratio  of  3  to  12,  by  referring  to  1  as  a  stand 
ard,  we  have 

3     :      12     :     :      1      :     ratio, 

which  brings  the  result  in  the  right  place. 

But  if  we  define  ratio  to  be  the  antecedent  divided  by  the  consequent, 
we  should  have 

U     :     12     :     :     ratio     :      I, 

wljitli  would  briiifi  the  ratio,  or  icqxivcd  number,  in  the  3J  place. 


RATIO    AND    PROPOKTIOIf. 


216 


What  are  the  ratios  of  the  proportions, 


3       : 

9 

:        12 

36? 

2       : 

10 

:       12 

:       60? 

4 

2 

:         8 

4? 

9       : 

1 

:       90 

10  ] 

225.  The  1st  and  4th  terms  of  a  proportion  are  caJl^r^  the 
extremes  t  the  2d  and  3d  terms,  the  means.  Thus,  in  th^  pm 
portion, 

3     :     12     :     :     6     :     24 


Since  (Art.  224), 


3  and  24  are  the  extremes,  and  12  and  6  the  means: 
12_24 
3  "T' 

we  shall  have,  by  reducing  to  a  common  denominator, 
12x6     24x3 
~3x6~    6x3* 

But  since  the  fractions  are  equal,  and  have  the  same  deno- 
minators, their  numerators  must  be  equal,  viz  ; 

12x6  =  24x3;  that  is, 

In  any  proportion^  the  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

Thus,  in  the  proportions, 

1   :     6  :   :     2  :   12;  we  have   1x12=   2x6; 
4  :   12  :   :     8  :  24 ;     "       "     4x24=12x8. 

226.  Since,  in  any  proportion,  the  product  of  the  extremes 
u  equal  to  the  product  of  the  means,  it  follows  that. 

In  all  cases,  the  numerical  value  of  a  quantity  is  the  number  of  timep 
which  that  quantity  contains  an  assumed  standard,  called  its  U7iit  oj 
mcai  ure. 

If  we  would  find  that  numerical  value,  in  its  right  place,  we  must 
aay, 

standard  quantity     :     :      1     :     numerical  value  : 

but  if  we  take  the  other  method,  we  have 

quantity  stjtndard  numerical  value     :      1. 

which  brijiys  the  uuuieritfcil  value  in  the  wruntr  place. 


216  •  RATIO    ANP    PRUPOKTION. 

1st.  If  the  product  of  the  means  be  divided  by  one  of  ike 
extremes^  the  quotient  will  be  the  other  extreme. 

Thus,  in  the  pioportion 

3   :   12  :  :  6   :  24,  we  have    3x24  =  12x6; 

then,  if  72,  the  product  of  the  means,  be  divided  by  one  o* 
the  extremes,  3,  the  quotient  wi]]  be  the  other  extreme,  24: 
or,  if  the  product  be  divided  by  24,  the  quotient  will  be  3. 

2d.  If  the  product  of  the  extremes  be  divided  by  either  of 
the  means ^  the  quotient  will  be  the  other  mean. 

Thus,  if  3x24  =12x6  :rr  72  be  divided  by  12,  the  quotient 
will  be  6  ;  or  if  it  be  divided  by  6,  the  quotient  will  be  12. 

EXAMPLES. 

1.  The  first  three  terms  of  a  proportion  are  3,  9  and  12  : 
■what  is  the  fourth  term  1 

2.  The  first  three  terms  of  a  proportion  are  4,  16  and  15  : 
what  is  the  4th  term  '\ 

3.  The  first,  second,  and  fourth  terms  of  a  proportion  are 
6,  12   and  24  :   what  is  the  third  term  ? 

4.  The  second,  third,  and  fourth  terms  of  a  proportion  are 
9,  6   and  24  :  what  is  the  first  term  ? 

5.  The  first,  second  and  fourth  terms  are  9,  18  and  48  ; 
what  is  the  third  termi 

227.   Simple  and  Compound  Ratio. 

The  ratio  of  two  single  numbers  is  called  a  Simple  Ratio^ 
and  the  proportion  which  arises  from  the  equality  of  two  such 
ratios,  a  Simple  Proportion. 

225.  Which  are  thti  extremes  of  a  proportion  ?  Which  the  means  ? 
What  is  the  product  of  the  extremes  equal  to  1 

226.  If  the  product  of  the  means  be  divided  by  one  of  the  extroinee, 
what  will  the  quotient  be  1  If  the  product  of  the  means  be  divided  by 
either  extreme,  what  will  the  quotient  be  \ 

227.  What  is  a  simple  ratio  ]  What  is  the  proportion  called  whicli 
comes  from  the  equality  of  two  simple  ratios  1  What  is  a  compouiul 
ratio  \     AVLut  is  a  cuuqtound  j)r«>portion  ? 


RAIIO    AND    PKOPOK'nON.  217 

If  the  terms  of  one  ratio  be  multiplied  by  the  terms  of  an 
other,  antecedent  by  antecedent  and  consequent  by  conse- 
quent, the  ratio  oi  the  products  is  called  a  Compound  Ratio. 
Thus,  if  the  two  ratios 

3     :     6     and     4     :     12 
be  multiplied  together,  we  shall  have  the  compound  ratio 
3x4     :     6x12,  or  12     :     72; 

in  which   the  ratio  is  equal   to  the  product  of  the  simple 
ratios. 

A  proportion  formed  from  the  equality  of  two  compound 
ratios,  or  from  the  equality  of  a  compound  ratio  and  a  simple 
ratio,  is  called  a  Comjyound  Proportion, 

228.    What  part  one  number  is  of  another. 

When  the  standard,  or  antecedent,  is  greater  than  the 
number  which  it  measures,  the  ratio  is  a  proper  fraction, 
and  is  such  a  part  of  1,  as  the  number  measured  is  of  the 
standard. 

1,  What  part  of  12  is  3  '?  that  is,  what  part  of  the  stand 
ard  12,  is  3  ? 

T^=i ;  or. 
12     :     3     :     :     1     :     i; 

that  is,  the  number  measured  is  one-fourth  of  the  standard. 


2.  What  part  of  9  is  2  ? 

3.  What  part  of  16  i&^l. 

4.  What  part  of  100  is  201 


7.  3  is  what  part  of  12  ? 

8.  5  is  what  part  of  20 '? 

9.  8  is  what  part  of  56  ? 


5.  What  part  of  300  is  200  ?   j  10.  9  is  what  part  of  8  ? 

6.  What  part  of  36  is  144  ?      |  11.   12  is  what  part  of  132  '? 

Note. — The  standard  is  generally  preceded  by  the  word  of,  and 
in  comparins:  numbers,  may  be  named  second,  as  in  examples  7, 
8,  9,  10  and  11,  but  it  must  always  be  used  as  a  divisor,  and 
Bhould  be  placed  first  in  the  statement. 

228  When  the  standard  is  greater  than  the  consequent,  hew  may 
the  ratio  he  compared  ?  WTiat  part  is  3  of  11  H  of  1  !  Wliat  part  is 
4of2'      l2of:t'      Vol;')' 

8 


21S 


SINGLE    RULE    OF    1 HREE. 


SINGLE  RULE  OF  THREE. 

229.  The  Single  Rule  of  Three  is  an  application  of  the 
principle  of  simple  ratios.  Three  numbers  are  always  giveii 
and  a  fourth  required.  The  ratio  between  two  of  the  given 
numbers  is  the  same  as  that  between  the  third  and  the  required 
number. 

1.  If  3  yards  of  cloth  cost  $12,  what  will  6  yards  cost  at  the 
same  rate  ? 

Note. — We  shall  denote  the  required  term  »f  the  proportion  by 
the  letter  x. 


STATEMENT. 


6  :  :  12 


OPERATION. 


12 


Ans.  a;  — $24. 


Analysis. — The  condition,  "  at  the  same 
rate,"  requires  that  the  quaidity  3  yards 
must  have  the  same  ratio  to  the  quantity  6 
yards,  as  $12,  the  cost  of  3  yards,  to  x  dol- 
lars, the  cost  of  12  yards. 

Since  the  product  of  the  two  extremes  is 
equal  to  the  product  of  the  two  means,  (Art. 
225),  3  X  a;  =  6  X  12 ;  and  if  3  x  a;  =  6  x  12,  a; 
must  be  equal  to  this  product  divided  by  8 : 
that  is, 

The  4th  term  is  equal  to  the  product  of  the  second  and  third 
terms  divided  by  the  first. 

2.  If  56  dollars  will  buy  14  yards  of  broadcloth,  how  many 
yards,  at  the  same  rate,  can  be  bought  for  84  dollars  ? 

Analysis. — Fifty-six  dollars,  (being 
the  cost  of  14  yards  of  cloth),  has  the 
same  ratio  to  $84,  as  14  yards  has  to  the 
number  of  yards  which  $84  will  buy. 

N"oTB. — When  the  vertical  line  is  used, 
the  required  term,  (which  is  denoted  by 
a;),  is  written  on  the  left. 


STATEMENT. 

$        $  yd.  yd, 

56  :  84  :  :  14  :  a;. 


OPERATION. 


H 


$4 


21 


a;  =  21 


229.  What  is  the  Single  Rule  of  Three?  How  many  numbers  are 
given  ?  How  many  required  ?  What  ratio  exists  between  two  of  the 
given  numbers? 


SINGLE    KULE    OF    THKEK.  219 

230.  Hence,  we  have  the  following 

Rule  I.  Write  the  number  which  is  of  the  same  kind  with 
the  answer  for  the  third  term^  the  number  named  in  connection 
with  it  for  the  first  term,  and  the  remaining  number  for  the 
second  term. 

II.  Multiply  the  second  and  third  terms  together^  and  divide 
the  product  by  the  first  term :  Or, 

Multiply  the  third  term  by  the  ratio  of  the  frst  and  second. 

Notes. — 1.  If  the  first  and  second  terms  have  different  units, 
they  must  be  reduced  to  the  same  unit. 

2.  If  the  third  term  is  a  compound  denominate  number,  it  must 
be  reduced  to  its  smallest  unit. 

3.  The  preparation  of  the  terms,  and  writing  them  in  their  pro- 
per places,  is  called  the  statement. 

EXAMPLES, 

1.  KI  can  walk  84  miles  in  3  days,  how  far  can  1  walk  in 
11  days? 

2.  If  4  hats  cost  %\2,  what  will  be  the  cost  of  66  hats  at 
the  same  rate  ? 

3.  If  40  yards  of  cloth  cost  $170,  what  will  325  yards  cost 
at  the  same  rate? 

4.  If  240  sheep  produce  660  pounds  of  wool,  how  many 
pounds  will  be  obtained  from  1200  sheep  1 

6.  If  2  gallons  of  molasses  cost  65  cents,  what  will  3  hogs- 
heads cost  ? 

6.  If  a  man  travels  at  the  rate  of  210  miles  in  6  days,  how 
far  will  he  travel  in  a  year,  supposing  him  not  to  travel  on 
Sundays  ? 

7.  If  4  yards  of  cloth  cost  $13,  what  will  be  the  cost  of  o 
pieces,  each  containing  25  yards? 

8.  If  48  yards  of  cloth  cost  $67,25,  what  will  144  y^rdg 
cost  at  the  same  rate  ? 

9.  If  3  common  steps,  or  paces,  are  equal  to  2  yards,  how 
many  yards  are  there  in  160  paces  1 

10.  If  750  men  require  22500  rations  of  bread  for  a  moiilh, 
how  many  rations  will  a  garrLson  of  1200  men  require  ? 

235.  Give  the  rule  for  the  Ktatcmeut.  Give  the  rule  for  findin/>  the 
fourth  tenn 


220 


SINGLE   RULE   OF   THREE. 


11.  A  cistern  containing  200  gallons  is  filled  by  a  pipe 
which  discharges  3  gallons  in  5  mniutes  ;  but  the  cistern  has 
a  leak  which  empties  at  the  rate  of"  1  gallon  in  5  minuies. 
If  the  water  begins  to  run  in  when  the  cistern  is  empty,  how 
long  will  it  run  before  filling  the  cistern  1 

12.  If  14^  yards  of  cloth  cost  $19^,  how  much  will  19|- 
yards  cost  ? 

Note, — First  make  the 
statement ;  then  change  the 
mixed  numbers  to  im- 
proper fractions,  after 
which  arrange  the  terms, 
and  cancel  equal  factors 
according  to  previous  in- 
struction. 


yard  of 


144 

X 

yd. 
19^ 

STATEMENT. 

yd.             $ 
191  :  :   19 

0.  4 

oth   cost  J 

2 
of  a  dollar, 

53  =  $26^ 
what  will 

13.  If  I  of  1 
2^  yards  cost  ? 

14.  If  YS  ^^  ^  ^^^^P  ^^^^  <£273  2s.  6d.,  what  will  ^^  of  her 
cost  1 

15.  If  ly*,  bushels  of  wheat  cost  $2|-,  how  much  will  60 
bushels  cost  1 

16.  If  4^  yards  of  cloth  cost  $9,75,  what  will  131  yardb 
cost? 

17.  If  a  post  8  feet  high  cast  a  shadow  12  feet  in  length, 
what  must  be  the  height  of  a  tree  that  casts  a  shadow  122 
feet  in  length,  at  the  same  time  of  day  ? 

18.  If  7cwt.  Iqr.  of  sugar  cost  $64,96,  what  will  be  the 
cost    uf  Acwt.  2qr.  1 

19.  A  merchant  failing  in  trade,  pays  65  cents  ibr  every 
dollar  which  he  owes:  he  owes  A  $2750,  and  B  $1975: 
how  much  does  he  pay  each  ? 

20.  If  6  sheep  cost  $15,  and  a  lamb  costs  one-third  as 
much  as  a  sheep,  what  will  27  lambs  cost  ? 

21.  If  2tbs.  of  beef  cost  -J-  of  a  dollar,  what  will  oOlbs. 
cost  ? 

22.  If  4^  gallons  of  molasses  cost  |2f ,  how  much  is  it  per 
quart  ? 

23.  A.  man  receives  |  oi'  his  income,  and  finds  it  equal  to 
$3724,10  :   Jjuw  iiiuch^is  his  whole  incuine  i 


BINGLE    KULE   OK    TKiiKK.  221 

24.  If  4  barrels  of  flour  cost  $34|,  how  much  can  be 
bought  for  $175^? 

25.  If  2  gallons  of  molasses  cost  65  cents,  what  will  3 
hogsheads  cost  ? 

26.  What  is  the  cost  of  6  bushels  of  coal  at  the  rate  of 
£1  145.  6d,  a  chaldron  '? 

27.  What  quantity  of  corn  can  I  buy  for  90  guineas,  at  the 
rate  of  6  shillings  a  bushel  ? 

28.  A  merchant  failing  in  trade  owes  ^3500,  and  his 
efiects  are  sold  for  $2100  :  how  much  does  B.  receive,  to 
whom  he  owes  $420  1 

29.  If  3  yards  of  broadcloth  cost  as  much  as  4  yards  of 
cassimere,  how  much  cassimere  can  be  bought  for  18  yards 
of  broadcloth] 

30.  If  7  hats  cost  as  much  as  25  pair  of  gloves,  .worth  84 
cents  a  pair,  how  many  hats  can  be  purchased  for  $216] 

31.  How  many  barrels  of  apples  can  be  bought  for  $1 14,33, 
if  7  ban-els  cost  $21,63] 

32.  If  27  pounds  of  butter  will  buy  45  pounds  of  sugar, 
how  much  butter  will  buy  36  pounds  ol"  sugar  ] 

33.  If  42^  tons  of  coal  cost  $206,21,  what  will  be  the  cost 
of  21  tons  ] 

34.  If  40  gallons  run  into  a  cistern,  holding  700  gallons,  in 
an  hour,  and  15  run  out,  in  what  time  will  it  be  filled  ] 

35.  A  piece  of  land  of  a  certain  length  and  12^  rods  in 
width,  contains  1^  acres,  how  much  would  there  be  in  a  piece 
of  the  same  length  26|  rods  wide] 

36.  If  13  men  can  be  boarded  1  week  for  $39,585,  what 
will  it  cost  to  board  3  men  and  6  women  the  same  time,  the 
women  being  boarded  at  half  price  ] 

37.  What  will  75  bushels  of  wheat  cost,  if  4 '  bushels  3 
pecks  cost  $10,687] 

38.  What  will  be  the  cost,  in  United  States  money,  of  324 
yards  oqjs.  of  cloth,  at  5ti.  46/.  New  York  currency,  for  2 
yards? 

39.  At  $1,12-|  a  square  foot,  what  will  it  cost  t(j  pav<»  a 
floor  18  fttet  long  and  12/^.  Oin.  wide] 


222  CAUSE    AND    EFFECT. 

CAUSE  AND  EFFECT. 

231.  Whatever  produces  effects^  as  men  at  work,  animals 
eating,  time,  goods  purchased  or  sold,  money  lent,  and  the 
like,  may  be  regarded  as  causes. 

Causes  are  of  two  kinds,  simple  and  compound. 

A  simple  cause  has  but  a  single  element,  as  men  at  work,  a 
portion  of  time,  goods  purchased  or  sold,  and  the  like. 

A  compound  cause  is  made  up  of  two  or  more  simple  ele- 
ments, such  as  men  at  work  taken  in  connection  tvith  time,  and 
the  like. 

232.  The  results  of  causes,  as  work  done,  provisions  con- 
sumed, money  paid,  cost  of  goods,  and  the  like,  may  be  re- 
garded as  effects.  A  simple  effect  is  one  which  has  but  a 
single  elttraent ;  a  compound  effect  is  one  which  arises  from 
the  multiplication  of  two  or  more  elements. 

233.  Causes  which  are  of  the  same  kind,  that  is,  which  can 
be  reduced  to  the  same  unit,  may  be  compared  with  each 
other ;  and  effects  which  are  of  the  same  kind  may  likewise 
be  compared  with  each  other.  From  the  nature  of  causes  and 
effects,  we  know  that 

1st  Cause  :  2d  Cause  :  :  1st  Effect  :  2d  Effect; 
and,  1st  Effect  ;  2d  Effect  :  :  1st  Cause  :  2d  Cause. 

234.  Simple  causes  and  simple  effects  give  rise  to  simple 
ratios.  Compound  causes  or  compound  effects  give  rise  to 
compound  ratios. 

Note. — Professor  H.  N.  Eobinson,  author  of  a  complete  course  of  Diathemaiics, 
fir.>t  made  a  practical  application  of  the  terms  "  Cause  and  Effect,"  in  the  development 
of  proportion,  as  published  in  his  arithmetic.  By  his  permission,  I  have  used  the 
same  terms,  but  have  somewhat  varied  the  method  and  rule. 

231.  Wlfat  are  causes?  How  many  kinds  of  causes  are  there? 
What  is  a  simple  cause?     What  is  a  compound  cause? 

2o2.  What  are  effects?  What  is  a  simple  effect?  What  is  a  com- 
pound effect? 

233  What  causes  are  of  the  same  kind?  What  causes  maybe  com- 
pared with  each  other?  What  do  we  infer  from  the  nature  of  causes 
and  effects  ? 

234.  What  gives  rise  to  simple  ratios? 


DOUBLE    RULE   OF    THREE.  223 


DOUBLE  RULE  OF  THREE. 

236.  The  Double  Rule  of  Tliree  is  an  application  of  the 
principles  of  compound  proportion.  It  embraces  all  that  class 
of"  questions  in  which  the  causes  are  compound,  or  in  which 
the  efiects  are  compound  ;  and  is  divided  into  two  parts  : 

1st.  When  the  compound  causes  produce  the  same  efiects; 
2d.  When  the  compound  causes  produce  different  effects. 

237.  When  the  cor/ipound  causes  produce  the  same  effects. 

1.  If  6  men  can  dig  a  ditch  in  40  days,  what  time  will  30 
men  require  to  dig  the  same  1 

Analysis. — The  first  cause 
is  compounded  of  6  men,  and 
40  days,  the  time  required  to 
do  the  work,  and  is  equal  to 
what  1  man  would  do  in 
6X40  =  240  days. 

The  second  cause  is  com- 
pounded of  30  men  and  the 
number  of  days  necessary  to 
do     the    same    work,     viz  : 

But  since  the  effects  are  the  ^ 

same,  viz  :  the  work  done,  the  causes  must  be  equal ;  hence,  the 
products  of  the  elements  of  the  causes  are  equal.  Therefore,  in  the 
solution  of  all  like  examples, 

Write  the  cause  containing  the  unknown  element  on  the  left 
of  the  vertical  line  for  a  divisor^  and  the  other  cause  on  the 
right  for  a  dividend. 

Note— This  class  of  questions  has  genera  iy  been  arranged 
under  the  head  of  "  Rule  of  Three  Inverse." 

EXAMPLES. 

1.  A  certain  work  can  be  done  in  12  days,  by  working  4 
hours  a  day  :  how  many  days  would  it  require  the  same 
number  of  men  to  do  the  same  work,  if  they  worked  6  hours 
a  day? 

236.  What  is  the  double  Rule  of  Three  1  What  cla«s  of  oue.'stions 
does  it  embrace  \     Into  how  many  parts  is  it  divided  1     ^^"hat  are  they  1 

237.  What  is  the  rule  when  the  effects  are  equal  \  LiiUer  what  rule 
has  this  class  of  cases  been  arranged  \ 


STATEMENT. 

men.   men.  \         ditch,     ditch, 
6    :   30     1     .  .     J    .      J 

days.  days.  \     '  ' 
40    :    X 

240    :    30x«    :    :    I     :    1. 

$0 

X 

0^ 

ii2'l  DOUBLE    RULE   OF   THKEE. 

2.  A  pasture  of  a  certain  extent  supplies  30  horses  for  18 
days  :  how  long  will  the  same  pasture  supply  20  horses  ? 

3.  If  a  certain  quantity  of  food  will  subsist  a  family  of  12 
persons  48  days,  how  long  will  the  same  food  subsist  a  family 
of  8  persons  ? 

4.  If  30  barrels  of  flour  will  subsist  100  men  for  40  days, 
how  long  will  it  subsist  25  men  ? 

5.  If  90  bushels  of  oats  will  feed  40  horses  for  six  days, 
how  many  horses  would  consume  the  same  in  1 2  days  1 

6.  If  a  man  perform  a  journey  of  22^  days,  when  the  days 
are  12  hours  long,  how  many  days  will  it  take  him  to  per- 
form the  same  journey  when  the  days  are  15  hours  long? 

7.  If  a  person  drinks  20  bottles  of  wine  per  month  when  it 
costs  2,s'.  per  bottle,  how  much  must  he  drink  without  increas- 
ing the  expense  when  it  costs  2s.  6d.  per  bottle  ] 

8.  If  9  men  in  18  days  will  cut  150  acres  of  grass,  how 
many  men  will  cut  the  same  in  27  days  ? 

9.  If  a  garrison  of  536  men  have  provisions  for  326  days, 
how  long  will  those  provisions  last  ii'  the  garrison  be  increased 
to  1304  men? 

10.  A  pasture  of  a  certain  extent  having  supplied  a  body 
of  horse,  consisting  of  3000,  wiih  ibrage  for  J 8  days  :  how 
many  days  would  the  same  pasture  have  supplied  a  body  of 
2000  horse  ? 

11.  What  length  must  be  cut  off  from  a  board  that  is  9 
inches  wide,  to  make  a  square  foot,  that  is,  as  much  as  is 
contained  in  12  inches  in  length  and  12  in  breadth "? 

12.  If  a  certain  sum  of  money  will  buy  40  bushels  of  oats 
at  45  cents  a  bushel,  how  many  bushels  of  barley  will  the 
same  money  buy  at  72  cents  a  bushel  ? 

13.  If  30  barrels  of  flour  will  support  100  men  for  40 
days,  how  long  would  it  subsist  400  men  1 

14.  The  governor  of  a  besieged  place  has  provisions  for  54 
days,  at  the  rate  of  2lb.  of  bread  per  day,  but  is  desirous  of 
prolonging  the  siege  to  80  days  in  expectation  of  succor  :  what 
muijt  h<i  the  ration  of  brea(J  i 


DOUBJ.E  KDLE  OF  THREE. 


225 


2?8.  When  the  Cornpound  Causes  'produce  different 
Effects. 

In  this  class  of  questions,  either  a  cause,  or  a  single  ele- 
ment of  a  cause  may  be  required  ;  or  an  effect,  or  a  single 
element  of  an  eflect  may  be  required. 

1.  If  a  family  of  6  persons  expend  $300  in  8  months,  how 
much  will  serve  a  lamily  of  15  persons  for  20  months  ? 

Analysis. — In    this    example    the    second  operation-. 

effect  is  required  ;  and  tlie  statement  may  be 
read  thus  ;  U  6  peisons  in  8  months  expend 
$300,  15  persons  in  20  months  will  expend 
how  many  (or  x)  dollars  ? 


^ 

15 

$ 

^0   ' 

X 

$00 

25 


2:=  1875  Am 


STATEMENT. 


1st  Cause  :  2d  Cause  :   :   1st  Effect  :  2d  Efle<  t. 


6 

8 

Or,  6x8 


15 

20 

15x20 


:  :  $300 


300 


x; 


2.   If  16  men,  in  12  days,  build  18  ieet  of  wall,  hov*^  man) 
men  must  be  employed  to  build  72  ieet  in  8  days  ] 

Analysis. — In  this  example  an  element  of 
the  second  cause  is  required,  viz :  the  number 
of  men.  The  question  may  be  read  thus : 
If  16  men,  in  12  days,  build  18  teet  of  wall, 
how  many  (or  x)  men,  m  8  days,  will  build 
72  feet  of  wall  ? 


1$ 


OPERATION. 

12 


ic=:96  days. 


STATEMENT. 


16 
12 

Or,     16X12 


X 

8 
xx8 


:     18     :     72; 
:     18     :     72. 


3.  If  32  men  build  a  wall  36  feet  long,  8  feet  high,  and 
4  feet  thick,  in  4  days,  working  12  hours  a  day;  how  long 
a  wall,  that  is  6  feet  high,  and  3  feet  thick  can  48  men  build 
in  36  days,  working  9  hours  a  day  ] 


238.  When  the  compound  causes  produce  different  effects,  what  will 
always  be  reiiuired  ' 
li. 


226 


DOUBLE    RULE    OF   THREE. 


Analysis. — In  this  example  an  element  of  the 
second  effect  is  required,  viz  :  the  length  of  the 
wall,  and  the  quet^tion  may  be  read  thus:  If 
32  men,  in  4  days,  working  12  hours  a  day, 
can  build  a  wail  36  feet  long,  8  feel  high,  and 
4  feet  thick,  48  men  in  36  days,  working  9 
hours  a  day,  can  build  a  wall  how  many  (or  x) 
feet  long,  6  feet  high,  and  3  feet  thick  ? 


OPERATION. 


4 

n 


36 


4 


2;=:  648  feet. 


STATEMENT. 

32  )  48  )  36  ) 

4^      :     set      :  :       8  V 

12)  9)  4) 

Or,  32x4x12  :  48x36x9   :   :  36x8x4 


% 
6 
3 

a;x6x3. 


Arrange  the  terms  in  the  statement  so  that  tht. 


239.   Hence,  we  have  the  following 

Rule. — I 
causes  shall  compose  one  couplet,  and  the  effects  the  other ^ 
fitting  X  in  the  place  of  the  required  element : 

II.  Then  if  x  fall  in  one  of  the  extremes,  make  the 
product  of  the  means  a  dividend,  and  the  product  of  the 
extremes  a  divisor  ;  hut  if  x  fall  in  one  of  the  means,  make 
the  pi'odMCt  of  the  extremes  a  dividend,  and  the  product  oj 
the  m>eans  a  divisor, 

EXAMPLES. 

1.  If  I  pay  |24  for  the  transportation  of  96  barrels  of  flour 
200  miles,  v/hat  must  I  pay  for  the  transportation  of  480  bar- 
rels 75  miles  ? 

2.  If  12  ounces  of  wool  be  sufficient  to  make  Ij  yards  ol 
cloth  6  quarters  wide,  what  number  of  pounds  will  be  required 
to  make  450  yards  of  flannel  4  quarters  wide  ? 

3.  V/hat  will  be  the  wages  of  9  men  for  11  days,  if  the 
wages  of  6  men  for  14  days  be  $84 '] 

4.  How  long  would  406  bushels  of  oats  last  7  horses,  if  154 
bushels  serve  14  horses  44  days? 

6.  If  a  man  travel  217  miles  in  7  days,  travelling  6  hours 
a  day,  how  ikr  would  he  travel  in  9  days  if  he  travelled  1 1 
hours  a  day  ? 


239.   What  is  the  rule  for  finding  the  unknown  part 


DOUBLJi.   RUhli   OF   THKEIi:.  '  227 

6.  If  27  men  can  mow  20  acres  of  grass  in  5J  days,  work- 
ing 3|  hours  a  day,  how  many  acres  can  10  men  mow  in  4^ 
days,  by  working  8^  hours  a  day '? 

7.  How  long  will  it  take  5  men  to  earn  $11250,  if  25  men 
can  earn  $6250  in  2  years  ? 

8.  If  15  weavers,  by  working  10  hours  a  day  for  10  days, 
can  make  250  yards  of  cloth,  how  many  must  work  9  hours 
a  day  for  15  days  to  make  607 J  yards  ? 

9.  A  regiment  of  100  men  drank  20  dollars'  worth  of  wine 
at  30  cents  a  bottle  :  how  many  men,  drinking  at  the  same 
rate,  will  require  12  dollars'  worth  at  25  cents  a  bottle  1 

10.  If  a  Ibotman  travel  341  miles  in  7i^  days,  travelling 
12^  hours  each  day,  in  how  many  days,  travelling  10 J  hours 
a  day,  will  he  travel   155   miles'^ 

11.  If  25  persons  consume  300  bushels  of  corn  in  1  year, 
how  much  will  139  persons  consume  in  8  months,  at  the 
same  rate  'i 

12.  How  much  hay  will  32  horses  eat  in  120  days,  if  9C 
horses  eat  3 J  tons  in  7^  weeks  1 

13.  If  $2,45  will  pay  for  painting  a  surface  21  feet  long 
and  13i  feet  wide,  what  length  of  surface  that- is  lOj  feet 
wide,  can  be  painted  for  $31,72  ? 

14.  How  many  pounds  of  thread  will  it  require  to  make 
60  yards  of  3  quarters  wide,  if  7  pounds  make  14  yards 
6  quarters  wide  1 

15.  If  500  copies  of  a  book,  containing  210  pages,  require 
12  reams  of  paper,  how  much  paper  will  be  required  to  print 
1200  copies  of  a  book  of  280  pages  ? 

16.  If  a  cistern  17^  feet  long,  101  feet  wide,  and  13  feet 
deep,  hold  546  barrels  of  water,  how  many  barrels  will  a 
cistern  12  feet  long,  10  feet  wide,  and  7  feet  deep,  contain? 

17.  A  contractor  agreed  to  build  24  miles  of  railroad  in  8 
months,  and  for  this  purpose  employed  150  men.  At  the 
end  of  5  months  but  1 0  miles  of  the  road  were  built  :  iiow 
many  more  men  must  be  employed  to  finish  the  road  in  the 
time  agreed  upon  ? 

18.  If  336  men,  in  5  days  of  10  hours  each,  can  dig  a  trench 
of  5  degrees  of  hardness,  7  0  yards  long  3  wide  and  2  deep  : 
what  length  of  trench  of  6  degrees  of  hardness,  5  yards  wide 
and  3  yards  deep,  may  be  dug  by  240  men  in  0  days  of  12 
hours  each '] 


228  PAKTNEK8HIP. 


PARTNERSHIP. 

240.  Partnership  is  the  joining  together  of  two  or  more 
persons  in  trade,  with  an  agreement  to  share  the  profits  or 
losses. 

Partners  are  those  who  are  united  together  in  carrying 
on  business. 

Capital,  is  the  amount  of  money  employed  : 
Dividend  is  the  gain  or  profit  : 
Loss  is  the  opposite  of  profit : 

241.  The  Capital  or  Stock  is  the  cause  of  the  entire  profit : 
Each  man's  capital  is  the  cause  of  his  profit  : 

The  entire  profit  or  loss  is  the  effect  of  the  whole  capital  : 
Each  man's  profit  or  loss  is  the  effect  of  his  capital :  hence, 
Whole  Stock  :   Each  man's  Stock 
:  :  Whole  profit  or  less  :  Each  man's  profit  or  loss. 

EXAMPLES. 

1.  A.  and  B  buy  certain  goods  amounting  to  160  dollars,  of 
which  A  pays  90  dollars  and  B,  70  ;  they  gain  32  dollars  by 
the  purchase  :  what  is  each  one's  share  ? 


OPERATION. 


160   :  90   :   :  32  :  A's  share  ;  or,  x 


n  18 


ar=z|18. 


^  100   I    U    1. 
160  :  70   :   :  32  :  B's  share  ;  or,  ^      70   ^^ 


a:  =  $14. 


!Si40.  What  is  a  partnership  1  What  are  partners  1  What  is  capita 
or  slock  !     What  is  dividend  !     What  is  loss  ^ 

241.  What  is  the  cause  of  the  protit  !  What  is  the  cause  of  each 
man's  profit  ?  What  is  the  ellcct  of  the  whole  capital  !  What  is  the 
eflect  of  each  man's  capital  !  \\  hat  prooortion  exis^ts  between  cause? 
and  their  eliects  '      What  is  the  rule  .' 


(X)MP(>UND    PARTNEKSiriP.  229 

Hence,  the  following 

Rule. — An  the  whole  stock  in  to  each  man's  share,  no  is  the 
whole  guia  or  loss  to  each  mail's  share  of  the  gain  or  loss. 

EXAMPLES. 

1.  A  and  B  have  a  joint  stock  of  $2100,  of  which  A  owns 
^1800  and  B  $300  ;  they  gain  in  a  year  $1000  :  what  is 
each  one's  share  of  the  profits  1 

2.  A,  B  and  C  fit  out  a  ship  for  Liverpool.  A  contributes 
S3 200,  B  $5000,  and  C  $4500  ;  the  profits  of  the  voyage 
amount  to  $1905  :  what  is  the  portion  of  each  ? 

3.  Mr.  Wilson  agrees  to  put  in  5  dollars  as  olten  as  Mr. 
Jones  puts  in  7  ;  after  raising  their  capital  in  this  way,  they 
trade  for  1  year  and  find  their  profits  to  be  $3600  :  what  is 
the  share  of  each  ? 

4.  A,  B  and  C  make  up  a  capital  of  $20,000  ;  B  and  C 
each  contribute  twice  as  much  as  A  ;  but  A  is  to  receive  one- 
third  of  the  profits  for  extra  services  ;  at  the  end  of  the  year 
they  have  gained  $4000  :  what  is  each  to  receive  i 

5.  A,  B  and  C  agree  to  build  a  railroad  and  contribute 
$18000  of  capital,  of  which  B  pays  2  dollars  and  C,  3  dollars 
as  often  as  A  pa)s  1  dollar ;  they  lose  $2400  by  the  opera- 
tion :  what  is  the  loss  of  each  ? 

COMPOUND  PARTNERSHIP. 

242.    When  the  causes  ofprojit  or  loss  are  compound. 

"When  the  partners  employ  their  capital  for  different  periods 
of  time,  each  cause  of  profit  or  loss  is  compound,  being  made 
up  of  the  two  elements  of  capital  and  tirne.  The  product  of 
these  elements,  in  each  particular  case,  will  be  the  cause  of 
each  man's  gain  or  loss ;  and  their  sum  will  be  the  cause  of 
the  entire  gain  or  loss  :  hence,  to  find  each  share, 

Multiply/  each  mans  stock  by  the  time  he  contimted  it  in 
trade  ;  then  sag,  as  the  sum  of  the  products  is  to  each  pro  duct , 
80  is  the  whole  gain  or  loss  to  each  mam's  sfiare  of  the  gaiii  or 
loss. 

242.  When  is  the  cause  of  profit  or  loss  compound  1  What  are  the 
elt'iiients  of  the  compound  cause  !     What  i?  the  rule  hi  this  case  ! 


230  ooMPoum)  pabtjsekshu* 


EXAMPLES. 


1.  A  and  B  entered  into  partnersliip.  A  put  in  $840  foi  4 
months,  and  B,  $650  for  6  months ;  ihey  gained  $363  ;  what 
is  each  one's  share  ? 


OPERATION. 


A.  8840x4  =  3360 

B.  650x6  =  3900 


7260  :  I  nil  :  :  363  :     j 


3360  :  :  o^o  .     {  $168  A's. 
$195  B's. 


2.  A  puts  in  trade  $550  for  7  months  and  B  puts  in  $1625 
for  8  months  ;  they  make  a  profit  of  $337  :  what  is  the 
share  of  each  1 

3.  A  and  B  hire  a  pasture,  for  which  they  agree  to  pay 
$92,50.  A  pastures  12  horses  for  ^  weeks  and  B  11  horses 
for  7  weeks  :  what  portion  must  each  pay  ? 

4.  Four  traders  form  a  company.  A  puts  in  $400  for  5 
months ;  B  $600  for  7  months  ;  C  ^960  for  8  months ;  D 
$1200  for  9  months.  In  the  course  of  trade  they  lost  $750  : 
how  much  falls  to  the  share  of  each  ? 

5.  A,  B  and  C  contribute  to  a  capital  of  $15000  in  the 
following  manner  :  every  time  A  puts  in  3  dollars  B  puts  in 
$5  and  C,  $7.  A's  capital  remains  in  trade  1  year  ;  B's  Ij 
years  ;  and  C's  2^  years  ;  at  the  end  of  the  time  there  is  a 
profit  of  $15000  :  what  is  the  share  of  each  ? 

6.  A  commenced  business  January  1st,  with  a  capital  of 
$3400.  April  1st,  he  took  B  into  pai  Lnership,  with  a  capital 
of  $2600  ;  at  the  expiration  of  the  year  they  had  gained 
$750  :  what  is  each  one's  share  of  the  gain  ? 

7.  James  Fuller,  John  Brown  and  William  Dexter  formed 
a  partnership,  under  the  firm  of  Fuller,  Brown  &  Co.,  with  a 
capital  of  $20000  ;  of  whioh  Fuller  furnished  $6000,  Brown 
$5000,  and  Dexter  S9000.  At  the  expiration  of  4  months, 
Fuller  furnished  $2000  more  ;  at  the  expiration  of  6  months, 
Brown  furnished  $2500  more  ;  and  at  the  end  of  a  year  Dex- 
ter withdrew  $2000.  At  the  expiration  of  one  year  and  a 
half,  they  found  their  profits  amounted  to  $5400  :  what  was 
each  partner's  share  ? 


PERCENTAGE. 


281 


PERCENTAGE. 

243.  Percentage  is  an  allowance  made  by  the  hundred. 

The  base  of  percentage,  is  the  number  on  which  the  per- 
centage is  reckoned. 

Per  cent  means  by  the  hundred  :  thus,  1  per  cent  means 

1  for  every  hundred  ;  2  per  cent,  2  for  every  hundred  ;  3  per 
cent,  3  for  every  hundred,  &c.     The  allowances,  1  per  cent, 

2  per  cent,  3  per  cent,  &c.,  are   called  rates,  and  may  be 
expressed  decimally,  as  in  the  following 

TABLE. 


1  per  cent  is 

.01 

7  per  cent  is 

.07 

3  per  cent  is 

.03 

8  per  cent  is 

.08 

4  per  cent  is 

.04 

15  per  cent  is 

.15 

5  per  cent  is 

.05 

68  per  cent  is 

.68 

6  per  cent  is 

.06 

99  per  cent  is 

.99 

ALSO, 

100  per  cent  is  1.      :  for,  \^  is  equal  to  1. 
150  per  cent  is  1.50  :  for,  \^  is  equal  to  1.50 
130  per  cent  is  1.30  :  for,  Ifa  is  equal  to  1.30 
200  per  cent  is  2.       :  for,  \^  is  equal  to  2.00 
^  per  cent  is  .005  :  for,  T^-r-2  is  equal  to  .005 
^  per  cent  is  .035  :  for,  3^=.03  +  .005  =  .035 
5f  per  cent  is  .0575  :  for,  5^=:=. 05+  .075=  075 

examples. 

Write,  decimally,  SJ  per  cent ;  9  per  cent ;  6J  per  cent ; 
65J  per  cent ;  205  per  cent ;  327  per  cent. 

244.   To  find  the  percentage  of  any  number. 

1.  What  is  the  percentage  of  $320,  the  rate  being  5  per 
cent? 

243.  What  is  per  centage  "?  What  is  the  base  1  What  does  per  cent 
mean  ]  What  do  you  understand  by  3  per  cent  ]  What  is  the  rate,  ot 
rate  per  cent  ] 

244.  How  du  you  find  the  percentage  of  any  number  I 


232 


PERCENTAGE. 


Analysis. — The  .  ate  being  6  per  cent,  is  ex-         oferatiom. 
pressed  decimally  by  .05.     We  are  then  to  take  320 

05  of  the  base  (which  is  $320);   this  we   do   by  ,05 

multiplying  $320  by  .05. 

Hence,  to  tind  the  percentage  of  a  number, 


$'16,00  Atis 


Multiply  the  number  hy  the  rate  expressed  decimally^  and 
the  product  will  be  the  jjercentage. 

EXAMPLES. 

1.   What  is  the  percentage  of  $G57,  the  rate  being  4-J  per 
cent  ? 


Note. — W^hen  the  rate  cannot  be 
reduced  to  an  exact  decimal,  it  is  most 
cojivenient  to  multiply  by  the  fraction, 
and  then  by  that  part  of  the  rate  which 
is  expressed  in  exact  decimals. 


OPERATION. 

657 

219=^  per  cent. 
26281=4  per  cent. 


$28,47  =  4^  per  cent. 
P'ind  the  percentage  of  the  following  numbers : 


1.  2-^  per  cent  of  650  dollars. 

2.  3  per  cent  of  650  yards. 

3.  41  per  cent  of  875cwL 
61  per  cent  of  *?37,50. 
5|  per  cent  of  2704  miles. 
J  per  cent  of  1000  oxen. 


2|  per  cent  of  $376. 


10.  66|  per  cent  of  420  cows. 

11.  105  per  cent  of  850  tons. 

12.  116  percent  of  875/^. 

13.  241  per  cent  of  $875,12. 

14.  37i  per  cent  of  |!200. 

15.  33i  per  cent  of  $687,24. 

16.  87i  per  cent  of  $400. 

17.  621  per  cent  of  $600. 

18.  308  per  cent  of  $225,40. 


4. 

5. 

6. 

7. 

S,   2^*0  P^r  cent  of  860  sheep. 

9.   5f  per  cent  of  $327,33. 

19.  A  has  $852  deposited  in  the  bank,  and  wishes  to  draw 
out  5  per  cent  of  it :  how  much  must  he  draw  for  ? 

20.  A  merchant  has  1200  barrels  of  flour  :  he  shipped 
64  per  cent  of  it  and  sold  the  remainder  :  how  much  did  he 
Bell? 

21.  A  merchant  bought  1200  hogsheads  of  molasses.  Oa 
getting  it  into  his  store,  he  found  it  short  3J  per  cent  •  how 
many  hogsheads  were  wanting  ? 

22.  What  is  the  diflerence  between  5^  per  cent  of  $800 
and  6^  per  cent  of  $1050  V 


23.  Two  men  had  each  $240.  One  of  them  spends  14 
per  cent,  and  the  other  18^  per  cent :  how  many  dollars  more 
did  one  spend  than  the  other  ? 

24.  Aman  has  a  capital  of  $12500:  he  puts  15  per  cent 
of  it  in  State  Stocks  :  33^  per  cent  in  Railroad  Stocks,  and 
25  per  cent  in  bonds  and  mortgages  :  what  per  cent  has  he 
left,  and  what  is  its  value  ? 

25 .  A  farmer  raises  850  bushels  of  wheat :  he  agrees  to 
sell  18  per  cent  of  it  at  §1,25  a  bushel ;  50  per  cent  of  it  at 
$1,50  a  bushel,  and  the  remainder  at  $1,75  a  bushel  :  how 
much  does  he  receive  in  all "? 

245.    To  find  the  per  cent  which  one  number  is  of  another. 

1.  What  per  cent  of  $16  is  $4  ] 

Analysis. — The  question  is,  what  part  of  operation. 

$16  is  $4,  when  expres.sed  in  hundredths:  y^z=-i-zr:.25. 

The  standard  is  $16  (Art.  228) :   hence,  the       qj.  25  per  cent. 
part  is  ^=^=.25  j  therefore,  the  percent   is 
25 :  hence,  to  find  what  per  cent  one   number  is  of  another, 

Divide  by  the  sta^idard  or  base,  and  the  quotient,  reduced 
to  decinuds,  iv ill  express  the  rate  per  cent. 

Note. — The  standard  or  basse,  is  generally  preceded  by  the  word 
of. 

EXAMPLES. 

1.  What  per  cent  of  20  dollars  is  5  dollars'? 

2.  Forty  dollars  is  what  per  cent  of  eighty  dollars  ? 

3.  What  per  cent  of  200  dollars  is  80  dollars  ? 

4.  What  per  cent  of  1250  dollars  is  250  dollars? 

5.  What  per  cent  of  650  dollars  is  250  dollars  1 

6.  Ninety  bushels  of  wheat  is  what  per  cent  of  IQQObush.  ? 

7.  Nine  yards  of  cloth  is  what  per  cent  of  870  yards  ? 

8.  Forty-eight  head  of  cattle  are  what  per  cent  of  a  drove 
of  1600? 

9.  A  man  has  $550,  and  purchases  goods  to  the  amount 
of  $82,75  :  what  per  cent  of  his  money  does  he  expend  ? 

245.  How  do  you  find  the  per  cent  which  one  number  is  of  another  t 


234  pekcentagp:. 

10.  A  merchant  goes  to  New  York  with  $1500  ;  he  first 
tays  out  20  per  cent,  after  M'hich  he  expends  $660  :  \vhat 
per  cent  was  his  last  purchase  of  the  money  that  remained 
after  his  first  1 

11.  Out  of  a  cask  containing  300  gallons,  60  gallons  are 
drawn  :  what  per  cent  is  this  1 

12.  If  I  pay  $698,23  for  3  hogsheads  of  molasses  and  sell 
ihem  for  $837,996,  how  much  do  1  gam  per  cent  on  the 
money  laid  outl 

13.  A  man  purchased  a  farm  of  75  acres  at  $42,40  an 
acre.  He  afterwards  sold  the  same  farm  for  $3577,50  :  what 
was  his  gain  per  cent  on  the  purchase  money  1 

STOCK,  COMMISSION  AND  BROKERAGE. 

246.  A  Corporation  is  a  collection  of  persons  authorized 
by  law  to  do  business  together.  The  law  which  defines  theii 
rights  and  powers  is  called  a  Charter. 

Capital  or  Stock  is  the  money  paid  in  to  carry  on  the 
business  of  the  Corporation,  and  the  individuals  so  contributing 
are  called  Stockholders.  This  capital  is  divided  into  equal 
parts  called  Shares,  and  the  written  evidences  of  ownership 
are  called  Certificates. 

247.  When  the  United  States  Government,  or  any  of  the 
States,  borrows  money,  an  acknowledgment  is  given  to  the 
lender,  in  the  form  of  a  bond,  bearing  a  fixed  interest.  Such 
bonds  are  called  United  States  Stock,  or  State  Stock. 

The  par  value  of  stock  is  the  number  of  dollars  named  in 
each  share.  The  market  value  is  what  the  stock  brings  per 
nhare  when  sold  for  cash. 

If  the  market  value  is  above  the  par  value,  the  stock  is 
said  to  be  at  a  premium,  or  above  par  ;  but  if  the  market 
value  is  below  the  par  value,  it  is  said  to  be  at  a  discount,  or 
below  par. 

246.  What  is  a  corporation  1  What  is  a  charter  1  What  is  capital 
01  stock  1     What  are  shares  1 

217.  What  are  United  States  Stocks^  What  are  State  Stocks  1 
What  is  the  par  value  of  a  stock  1  What  is  the  market  value!  If  the 
market  is  above  the  par  value,  what  is  said  of  the  stock  1  If  it  is  below, 
what  is  said  of  the  stock  1  What  is  the  market  value  when  above  pari 
\Miat  when  below  1 


COMMISSION    AND    BKOXERACJB.  235 

Let  1  =  par  value  of  1  dollar  : 

I H- premium  ^market  value  of  1   dollar,  when  above 

par  : 
1 — discount  =:maket  value  of  I  dollar  when  below  par. 

248.  Commission  is  an  allowance  made  to  an  agent  for 
buying  or  selling,  and  is  generally  reckoned  at  a  certain  rate 
per  cent. 

The  commission,  for  the  purchase  or  sale  of  goods  in  the 
city  of  New  York,  varies  from  2^  to  1 2^  per  cent,  and  under 
some  circumstances  even  higher  rates  are  paid. 

Brokerage  is  an  allowance  made  to  an  agent  who  buys  or 
sells  stocks,  uncurrent  money,  or  bills  of  exchange,  and  is 
generally  reckoned  at  so  much  per  cent  on  the  par  value  of 
the  stock.  The  brokerage,  in  the  city  of  New  York,  is  gene- 
rally one-fourth  per  cent  on  the  par  value  of  the  stock. 

EXAMPLES. 

1.  What  is  the  commission  on  $4396  at  6  per  cent  1 

OPERATION. 

Note.  —  We   here   find   the   commission,  as  $4396 

in  simple  percentage,  by  multiplying  by  the  de-  06 

cimal  which  expresses  the  rate  per  cent.  .        . -,^  '„, 

^  ^  Ans.  $263,76. 

2.  A  factor  sells  60  bales  of  cotton  at  $425  per  bale,  and 
is  to  receive  2^  per  cent  commission :  how  much  must  he  pay 
over  to  his  principal  ? 

3.  A  drover  agrees  to  purchase  a  drove  of  cattle  and  to  sell 
them  in  New  York  city  for  5  per  cent  on  what  he  may  re- 
ceive ;  he  expends  in  the  purchase  $4250,  and  sells  them  at 
an  advance  of  10  per  cent :  how  much  is  his  commission  % 

4.  A  commission  merchant  sells  goods  to  the  amount  ol 
$8750,  on  which  he  is  to  be  allowed  2  per  cent,  but  in  con- 
sideration of  paying  the  money  over  before  it  is  due,  he  is  to 
receive  1 J  per  cent  additional :  how  much  must  he  pay  ovei 
to  his  principal  ? 

5.  A  broken  bank  has  a  circulation  of  $98000  and  pur- 
chases the  bills  at  85  per  cent*:  how  much  is  made  by  the 
operation  '\ 


248.  What  is  coinmissiou  1     What  m  brokerape! 


236  TMCIMIONTAGE. 

6.  Merchant  A  sent  to  B,  a  broker,  $3825  to  be  invested  in 
stock  ;  B  is  to  receive  2  per  cent  on  the  amount  paid  for  the 
stock  :  what  was  the  value  of  the  stock  purchased  1 


Analysis. — Since  the  broker  re- 
seives  2  per  cent,  it  will  require 
fl.02  to  purchase  1  dollar's  worth 
of  stock;   hence,   there  will   be  as  765 

many    dollars    worth    purchased   as  714 

$1.02   is  contained  times  in  $3825 
that  is,  $3750  worth. 


OPERATION. 

1.02)3825.00($3750Jn». 
306 


510 
510 


7.  Mr.'  Jones  sends  his  broker  $18560  to  be  invested  in 
U.  S.  Stocks,  which  are  15  per  cent  above  par  ;  the  broker  is 
to  receive  one  per  cent  ;  how  many  shares  of  $100  each  can 
be  purchased  ? 

Analysis. — Since  the  premium  is  15 

per  cent,  and   the  brokerage  1  per  cent,  operation. 

each  dollar  of  par  value  will  cost  $1  1.16)18560 

plus  the  premium  plus  the  brokerage^  "     ^TTT.T^  nnntlpnt 

$1.16:  hence,   the   amount    purchased  ^IbUUU  quotient, 

will  be   as  many  dollars  as  $1.16   is  or,      IbO        shares, 
contained  times  in  $18560. 

8.  I  have  $5000,89  to  be  laid  out  in  stocks,  which  are  15 
per  cent  below  par  :  allowing  2  per  cent  commission,  how 
much  can  be  purchased  at  the  par  value  1 

Analysis. — Since   the  stock   is  at  a  dis- 
count of  15  per  cent,  the  market  value  will  operation. 
be  85  per  cent;  add  2  per  cent,  the  broker-  .87)5000,89 

age.    gives  87  per  cent=.S7.     The  amount  ^5747 Ans 

purchased  will  be  as  many  dollars  as  .87  is  ^          ' 
contained  times  in  $5000,89. 

Hence,  to  find  the  amount  at  par  value, 

Divide  the  amount  to  be  expended  by  the  market  value  of 
$1  plus  the  brokerage  ];  and  the  quotient  will  be  the  amount 
in  par  value. 

9.  Messrs.  Sherman  &  Co.  rfeceive  of  Mr.  Gilbert  $28638,50 
to  be  invested  in  bank  stocks,  which  are  121  per  cent  above 
par,  for  which  they  are  to  receive  one-fourth  of  one  per  cent 
commission  :  how  many  shares  of  $127  each  can  they  buy  1 


LOSS   Oii   GAIN.  237 

10.  The  par  value  of  Illinois  Railroad  stock  is  100.  It 
Eells  in  market  at  72^:  if  I  pay  |  per  cent  brokerage,  how 
many  shares  can  I  buy  for  $5820  ? 

PROFIT  AND   LOSS. 

249.  Profit  or  loss  is  a  process  by  which  merchants  dis- 
cover the  amount  gained  or  lost  in  the  purchase  and  sale  of 
goods.  It  also  instructs  them  how  much  to  increase  or 
diminish  the  price  of  their  goods,  so  as  to  make  or  lose  so 
much  per  cent. 

EXAMPLES. 

1.  Bought  a  piece  of  cloth  containing  Ibyd.  at  $5,25  per 
yard,  and  sold  it  at  $5,75  per  yard  :  how  much  was  gained 
in  the  trade  % 

OPERATION. 

Analysis. — We    first   find    the      $5,75  price  of  1  yard. 
profit  on  a  single   yard,  and  then       $5,25  cost  oi"  1  yard. 

multiply  by  the  number  of  yards,       77- ,  r^        ■,  ^ 

which  is  75.  ^'^^'^^■-  P>-o^»t  on  1  yard  : 

then,  $0,50x75:::=  $37,50. 

2.  Bought  a  piece  of  calico  containing  56  yards,  at  27  cents 
a  yard  :  what  must  it  be  sold  ibr  per  yard  to  gain  $2,24  ? 

OPERATION. 

56  yards  at  27  cents  ==$15, 12 
An.alysis. — First    find    the      Profit  -  -  -  2,24 

cost,  then  add    ilie   profit  and      tx  ^      n  /•  7  ■,  ->  ^n' 

divide  the  sum  by  the  number       It  must  sellfor     -  $17,36. 

of  yards.  56)17,36 

31  cents. 

250.  Knowing  the  per  cent  of  gain  or  loss  and  tlie 
amount  received,  to  find  the  cost. 

1.  1  sold  a  parcel  of  goods  for  $195,50,  on  which  I  made 

15  per  cent :  what  did  they  cost  me  ? 

Analysis. — 1  dollar  of  the  cost  plus  \6  per  operation. 

cent,  will  be  what  that  which  cost  Si  sold  lor,  1.15)195,50 
viz.,   Si, 15  :  hence,    there   will  be    as    many  ^Tto — /I 

dollars  of  cot«t,  as  $1.15  is  contained  times  in  %  J      . 

what  the  goods  brought. 


'249.  What  is  loss  or 


gain 


238  I'EKCEWTAGE. 

2.  [[  I  sell  a  parcel  of  jroods  for  ^170,  by  which  1  losc 
15  per  cent,  what  did  they  cost  I 

Analysis. — 1  dollar  of  the  cost  less    15   per  operatiok. 

cent,  will  be  what  that  which  cost  1  dollar  sold         .85)170 
for,  viz.,  $0,85  :  hence,  there  will  be  as   many  skQrul"   A 

dollars  of  cost,  as  .85   is  contained    times   in  ^     . 

what  the  goods  brought. 

Hence,  to  find  the  cost, 

Divide  the  aniou?tt  received  by  1  jplus  the  per  cent  when 
there  is  a  gain,  and  by  1  minus  the  per  cent  when  there 
is  a  loss,  and  the  quotient  will  be  the  cost. 

EXAMPLES. 

1.  Bought  a  piece  of  cassimere  containing  28  yards  at 
IJ  dollars  a  yard  ;  but  finding  it  damaged,  am  f^'illing  to  sell 
it  at  a  loss  of  15  per  cent:  how  much  must  be  asked  per 
yard  ? 

2.  Bought  a  hogshead  of  brandy  at  $1,25  per  gallon,  and 
Bold  it  for  |78  :  was  there  a  loss  or  gain  ? 

3.  A  merchant  purchased  3275  bushels  of  wheat  for  which 
he  paid  |3517,10,  but  finding  it  damaged,  is  willing  to  lose 
10  per  cent :  what  must  it  sell  for  per  bushel? 

4.  Bought  a  quantity  of  wine  at  $1,25  per  gallon,  but  it 
proves  to  be  bad  and  am  obliged  to  sell  it  at  20  per  cent  less 
than  I  gave  :  how  much  must  1  sell  it  for  per  gallon  ? 

5.  A  farmer  sells  125  bushels  of  corn  for  75  cents  per 
bushel ;  the  purchaser  sells  it  at  an  advance  of  20  per  cent : 
how  much  did  he  receive  for  the  corn  ? 

6.  A  merchant  buys  1  tun  of  wine  for  which  he  pays  $725, 
and  wishes  to  sell  it  by  the  hogshead  at  an  advance  of  15  per 
cent  :  what  must  be  charged  per  hogshead  ? 

7.  A  merchant  buys  158  yards  of  calico  for  which  he  pays 
20  cents  per  yard  ;  one-half  is  so  damaged  that  he  is  obliged 
to  sell  it  at  a  loss  of  6  per  cent :  the  remainder  he  sells  at  aii 
advance  of  19  per  cent  :  how  much  did  he  gain  ? 

8.  If  I  buy  coflee  at  1 6  cents  and  sell  it  at  20  cents  a 
pound,  how  much  do  I  make  per  cent  on  the  money  paid  '\ 

250.  Knowing  the  per  cent  of  gain  or  loss  and  the  amount  received, 
£io\v  do  jou  find  the  cost  ' 


IN8UKAN0E.  289 

9  A  man  bought  a  house  and  lot  for  '^1850,50,  and  sold  it 
for  $1517,41  :  how  much  per  cent  did  he  lose.  ? 

10.  A  merchant  bought  650  pounds  oi'  cheese  at  10  cents 
per  pound,  and  sold  it  at  12  cents  per  pound  :  how  much  did 
he  gain  on  the  whole,  and  how  much  per  cent  on  the  money 
laid  out  ? 

11.  Bought  cloth  at  $1,25  per  yard,  which  proving  bad,  I 
wish  to  sell  it  at  a  loss  of  18  per  cent:  how  much  must  1 
ask  per  yard  ? 

12.  Bought  50  gallons  of  molasses  at  75  cents  a  gallon, 
10  gallons  of  which  leaked  out.  At  what  price  per  gallon 
must  the  remainder  be  sold  that  I  may  clear  10  per  cent  on 
the  cost  ? 

13.  Bought  67  yards  of  cloth  for  $112,  but  19  yards  being 
spoiled,  I  am  willing  to  lose  5  per  cent :  how  much  must  1 
sell  it  for  per  yard  ? 

14.  Bought  67  yards  of  cloth  for  $112,  but  a  number  of 
yards  being  spoiled,  I  sell  the  remainder  at  $2,21 6J  per  yard, 
and  lose  5  per  cent  :  how  many  yards  were  spoiled  ? 

15.  Bought  2000  bushels  of  wheat  at  $1,75  a  bushel,  from 
which  was  manufactured  475  barrels  of  flour  :  what  must 
the  flour  sell  for  per  barrel  to  gain  25  per  cent  on  the  cost  of 
the  wheat? 

INSURANCE. 

251.  Insurance  is  an  agreement,  generally  in  writing,  by 
which  an  individual  or  company  bind  themselves  to  exempt 
the  owners  of  certain  property,  such  as  ships,  goods,  houses, 
&c.,  from  loss  or  hazard. 

The  Policy  is  the  written  agreement  made  by  the  parties. 

Premium  is  the  amount  paid  by  him  who  owns  the  property 
to  those  who  insure  it,  as  a  compensation  for  their  risk.  The 
premium  is  generally  so  much  per  cent  on  the  prop»ity  in- 
sured. 

EXAMPLES. 

1.  What  would  be  the  premium  for  the  insurance  of  a 
house  valued  at  $8754  against  loss  by  fire  for  one  year,  at 
J  per  cent? 

251.  What  is  insurance  1  What  is  the  policy  ?  What  is  the  pre- 
luiuui  1     How  is  it  reckoned  ? 


240  PEllOENTAGE. 

2.  What  Avauld  be  the  premium  tor  insuring  a  ship  anJ 
cargo,  valued  at  ^37500,  liom  New  York  to  Liverpool,  at  3^ 
per  cent  ? 

3.  What  would  be  the  insurance  on  a  ship  valued  at 
$47520  at  ^  per  cent  ;  also  at  ^  per  cent? 

4.  What  would  be  the  insurance  on  a  house  valued  at 
$14000  at   1^  per  cent? 

5.  What  is  the  insurance  on  a  store  and  goods  valued  at 
$27000,  at  2J  percent? 

6.  What  is  the  premium  of  insurance  on  $9870  at  14  per 
cent? 

7.  A  merchant  w^ishes  to  insure  on  a  vessel  and  cargo  at 
sea,  valued  at  $28800  :  what  will  be  the  premium  at  Ij  per 
cent  ? 

8.  A  merchant  owns  three-fourths  of  a  ship  valued  at 
$24000,  and  insures  his  interest  at  2^  per  cent  :  what  does 
he  pay  for  his  policy  ? 

9.  A  merchant  learns  that  his  vessel  and  cargo,  valued 
at  S36000,  have  been  injured  to  the  amount  of  $12000  ;  he 
eflects  an  insurance  on  the  remainder  at  5^  per  cent ;  what 
premium  does  he  pay  ? 

10.  My  furniture,  worth  $3440,  is  insured  at  2|  per  cent ; 
my  house,  worth  $1000,  at  1^  per  cent  ;  and  my  barn,  horses 
and  carriages,  worth  $1500,  at  3;J-  percent:  what  is  the 
whole  amount  of  my  insurance  ? 

11.  A  man  bought  a  house,  and  paid  the  insurance  at  2^ 
per  cent,  the  whole  of  which  amounted  to  $1845  :  what  was 
the  value  of  the  house  and  the  amount  of  the  insurance? 

12.  What  would  it  cost  to  insure  a  store,  worth  $3240,  at 
I  per  cent,  and  the  stock,  worth  $7515,75,  at  ^  per  cent  ? 

13.  A  merchant  imported  250  pieces  of  broadcloth,  each 
piece  containing  36^  yards,  at  $3,25  cents  a  yard.  He  paid 
4J  per  cent  insurance  on  the  selling  price.  $4,50  a  yard.  If 
the  goods  were  destroyed  by  fire,  and  he  got  the  amount  of 
insurance,  how  much  did  he  make  ? 

14.  A  vessel  and  cargo,  worth  $65000,  are  damaged  to  the 
amount  of  20  per  cent,  and  there  is  an  insurance  of  50  per 
cent  on  the  loss  :   how  much  will  the  owner  receive  ? 


INTEREST.  241 


INTEREST. 

252  Interesi  is  an  allowance  made  for  the  use  of  money 
that  is  borrowed. 

Phincipal  is  the  money  on  which  interest  is  paid. 

Amount  is  the  sum  of  the  Principal  and  Interest. 

For  example  :  If  I  borrow  1  dollar  of  Mr.  Wilson  for  1 
vear,  and  pay  him  7  cents  for  the  use  of  it ;  then, 

1  dollar  is  the  pinncipal, 
.   7  cents  is  the  interest,  and 
^1,07  the  amount 

The  RATE  of  interest  is  the  number  of  cents  paid  for  the 
use  of  1  dollar  for  1  year.  Thus,  in  the  above  example,  the 
rate  is  7  per  cent  per  annum. 

Note. — The  term  per  cent  meaus,  by  the  hundred;  and  fer 
annum  means  by  the  year.  As  interest  is  always  reckoned  by  the 
year,  the  term  per  annum  is  understood  and  omitted. 

CASE   I. 

253.    To  find  the  interest  of  any  principal  for  one  or  more 

years. 

1.  What  is  the  interest  of  $1960  for  4  years,  at  7  per 
cent  ? 

Analysis. — The    rate    of    interest 

being  7  per  cent,   is  expressed  deci-  operation. 

mally  by  .07  :  hence    each  dollar,  in  $1960 

1  year,  will  produce  .07  of  itself,  and  .07  rate. 

$1960    will    produce    .07   of   $1960,  ,o^  on  •   ^  V      i 

or  $137,20.    Therefore,  $137,20  is  the  1^7,20  int.  for  \yr. 

interest  for  1    year,  and  this  interest  j^  ^^-  ^^  y^ars. 

rauhiplied  by  4,  gives  the  interest  for  |548,80 
4  years  :  hence,  the  following 

Rule. — Multiply  the  pi-incipal  by  the  rate,  exjyressed 
decimally,  and  the  product  by  the  number  of  years. 

252.  What  is  interest  ■  What  is  principal  ^  What  is  amount  ? 
What  is  rate  of  interest  1     What  does  per  annum  mean  1 

253.  How  do  you  find  the  interest  of  any  principal  for  any  number  ot' 
years  ^      Tiivp  the  analy:-is 

10 


242  SIMPLE    INTJCliEBT. 

EXAMPLES. 

1.  What   is  the  interest  of  $365,874   for  one  year,  at  5j 
percent? 

OPERATION. 

Analysis.— We  first  find  the  in-  ^^^^'^^ti 

terest  at   i  per  cent,  and  then  the  '}_zJ 

interest  at  5  per  cent;  the  sum  is  1,82937  ^  per  cent, 

the  interest  at  5i  per  cent.  18,29370  5  per  cent. 

Ans.  12042307  5^  per  cent 

2.  What  is  the  interest  of  $650  for  one  year,  at  6  per  cent  ? 

3.  What  is  the  interest  of  $950  for  4  years,  at  7  per  cent  ? 

4.  What  is  the  amount  of  13675  in  3  years,  at  7  per  cent  ? 

5.  What  is  the  amount  of  $459  in  5  years,  at  8  per  cent? 

6.  What  is  the  amount  of  $375  in  2  years,  at  7  per  cent  ? 

7.  What  is  the  interest  of  $21 1,26  for  1  year,  at  4^  per  ct.  ? 

8.  What  is  the  interest  of  $1576,91  for  3  years,  at  7  per  ct.  ? 

9.  What  is  the  amount  of  $957,08  in  6  years,  at  31  per  ct.  ? 

10.  What  is  the  interest  of  $375,45  for  7  years,  at  7  per  ct.  ? 

11.  What  is  the  amount  of  $4049,87  in  2  years,  at  5  per  ct.  1 

12.  Whatisthe  amount  of  $16 199,48  in  16  yrs.,at  5^perct.? 

Note. — When  there  are  years  and  months,  and  the  months  are 
aliquot  parts  of  a  year,  multiply  the  interest  for  1  year  by  the  yean 
i»nd  months  reduced  to  the  fraction  of  a  year. 

EXAMPLES. 

1.  What   is   the   interest   of    $326,50,    for   4    yeais    and 

2  months,  at  7  per  cent '? 

2.  What    is   the    interest   of    $437,21,    for    9    years   and 

3  months,  at  3  per  cent  ? 

3.  What  is  the  amount  of  $1119,48,  after  2  years  and 
6  months,  at  7  per  cent  1 

4.  What    is    the    amount  of  $179,25,  after  3  years   and 

4  months,  at  7  per  cent  ? 

5.  What  is  the  amount  of  $1046,24,  after  4  years  and 
3  uiuulhs,  at  5J  per  cent? 


grMl'LlC    INTKRKST.  243 


CASE   II. 

254.  To  find  the  interest  on  a  given  principal  for  any  rate 
»nd  time. 

1.  What  is  the  interest  of  $876,48  at  6  per  cent,  for 
4  years  9  months  and  14  days? 

Analysis. — The  interest  for  1  year  is  the  product  of  the  jirinci- 
pal  rnujtiplied  by  the  rate.  If  the  interest  for  1  year  be  divided 
by  12,  the  quotient  will  be  the  interest  for  1  month  :  if  the  interest 
for  1  month  be  divided  by  30.  the  quotient  will  be  the  interest 
for  1  day. 

The  interest  for  4  years  is  4  times  the  interest  for  1  year :  the 
interest  for  9  months,  9  times  the  interest  for  1  month ;  and  the 
interest  for  14  days,  14  times  the  interest  for  1  day. 

OPERATION. 

$876,48 
.06 


12)52,5888r=int.  for  lyr.     52,5888    X    4=$210,3552  4yr. 
30)4^3824 rz: int.  for  1  mo.     4,3824    x    9  =  $  39,4416  9mo 
,14608=:int.  for  Ida.       ,14608  xl4r^^     2,.0451  Uda 
Total  interest,  $251,84424- 
Hence,  we  have  the  following 
Rule. —  I.    Find  the  interest  for  1  year: 

II.  Divide  this  interest  by  12,  and  the  quotient  will  be  the 
interest  for  1  month: 

III.  Divide  the  interest  for  1  month  by  30,  and  the  quo- 
tient will  be  the  interest  for  1  day. 

IV.  Multiply  the  interest  for  1  year  by  the  number  of 
years.,  the  interest  for  1  month  by  the  number  of  months.,  and 
the  interest  for  1  day  by  the  number  of  days,  and  the  sum 
f  the  products  will  be  the  required  interest. 

Note. — In  computing  interest  the  month  is  reckoned  at  30  days. 

2.  What  is  the  interest  of  $132,26  for  1  year  4  months 
..nd  10  days,  at  6  per  cent  per  annum? 

3.  W'hat  is  the  interest  of  825,50  for  1  year  9  months  and 
12  days,  at  0  per  cenf? 

2S4.   How  do  ywu  find  the  iiittrekt  for  uiiy  tunc  at  any  rule ' 


244  8IMPI.E    INTEKEfeT. 

2d  method. 

255.  There  is  another  rule  resulting  from  the  last  analysis, 
which  is  regarded  as  the  best  general  method  of  computing 
interest. 

Rule. — I.  Find  Ike  interesl  for  1  year  and  divide  it  by  12  : 
the  quotient  will  be  the  interest  for  1  month. 

II.  Multiply  the  interest  for  1  month  by  the  time  expressed 
in  months  and  parts  of  a  month^  and  the  product  will  be  the 
required  interest. 

Note. — Since  a  month  is  reckoned  at  30  days,  any  number  of 
days  is  reduced  to  decimals  of  a  month  by  dividing  the  days  by  3. 

EXAMPLES. 

1.  What  is  the  interest  of  $327,50  for  3  years  7  mouths 
and  13  days,  at  7  per  cent? 

OPERATION. 

3y/rs.  =  36wos.  $327,50 

Imos.  .07 

13  days- .^mos.  12)22.9250      zuint.  fbr  1  year. 

Time=z43.4^-mos.  1.91044- =int.  tor  1  month. 

Note.— Tiie   method  em-  43.41   —time  in  months. 

j)loyed,  and    the  number  of  .6308 

decimal  places  used,  in  com-  76416 

puting   interest,    may  affect  57312 

the   mills,  and  possibly,  the  nf,A-\a 

last  figure  in  cents.   It  is  best 

to  use  4  places  of  decimals.  $82.97504   Ans. 

2.  What    is  the  interest  of  ^1728,00,  at   7  per  cent,  for 

2  years  6  months  and  21  days'? 

3.  What  is  the   interest  of  $288,30,    at  7   per   cent,  for 

I  year  8  months  and  27  days  1 

4.  What  is   the  interest  of  $576,60,   at    6  per   cent,   for 
10  months  and  18  days  ? 

5.  What   is  the  interest  of  $854,42,    at   6    per   cent,    fbr 

3  months  and  9  days  ? 

6.  What  is  the   interest  of  $1153,20,  at  6  per    cent,  for 

I I  months  and  6  days  ? 

265.  How  do  you  find  the  interest  for  ycirb,  njoiitlis  and  (hiyt*  by  the 
sccund  lactliod  ! 


SIMPLE    i:^'TEKl!:ST.  2iL 

7.  "What  is  the   interest  of  $2306,54,  at  5  per   cent,   fot 
7  mouths  and  28  days'? 

8.  What   is  the  interest  of  $4272,10,  at  5   per  cent,   ibi 
1 0  months  and  28  days  ? 

9.  What  is  the  interest  of  $1620,  at  4  per  cent,  for  5  years 
lid  24  days  ? 

10.  What  is  the  interest  of  $2430,72,  at  4  per  cent,  for 
10  years  and  4  months  ? 

11.  What  is   the  interest   of  $3689,45,  at   7  per  cent,  for 
4  years  and  7  months  ? 

12.  What  is  the  interest  of  $2945,96,  at   7  per  cent,  for 
7  years  and  3  days  1 

13.  What  is  the  interest,  at  8  per  cent,  of  $675,89,  for 
b  years  6  months  and  6  days  ? 

14.  What  is  the  interest,  at  8  per  cent,   on  $12324,  for 

3  years  and  4  months  ? 

15.  What  is  the  interest,  at  9  per  cent,  on  $15328,20,  ibr 

4  years  and  7  months  ? 

16.  What  is  the  interest  of  $69450  for  1  year  2  montha 
and  12  days,  at  9  per  cent  ? 

17.  What  is  the  interest  of  $216,984  for  3  years  5  months 
and  15  days,  at  10  per  cent? 

18.  What  is  the  interest  of  $648,54  for  7  years  6  months, 
at  4i  per  cent  ? 

19.  What  is  the  interest  of  $1297,10  for  8  years  5  months, 
at  5  J  per  cent  ? 

20.  What  is  the  interest  of  $864,768  for  9  months  25  day?, 
at  6 J  per  cent? 

21.  What  is  the  interest  of  $2594,20  for  10  months  and  9 
days,  at  7^  per  cent  i 

22.  What  is  the  amount  of  $2376,84  for  3  years  9  months 
and  12  days,  at  8^  per  cent '? 

23.  What  is  the  amount  of  $5148,40  for  7  years  11  months 
and  23  days,  at  9^  per  cent  ? 

24.  What  is  the  amount  of  $3565,20  lor  3  years  9  months, 
at  10-J  [^»er  cent  ^ 


246  SIMPLE   INTEREST. 

25.  What  is  the  amount  of  $125,75  for  1  year  9  months 
and  27  days,  at  7  per  cent  1 

26.  What  is  the  amount  of  $256  for  10  months  15  days,  at 
7^  per  cent  ? 

27.  What  is  the  interest  on  a  note  of  $264,42,  given  Janu- 
ary 1st,  1852,  and  due  Oct.  10th,  1855.  at  4  per  cent? 

28.  Gave  a  note  of  $793,26  April  6th,  1850,  on  interest  at 
7  per  cent  :  what  is  due  September  10th,  1852  1 

29.  What  amount  is  due  on  a  note  of  hand  given  June  7th, 
1850,  for  $512,50,  at  6  per  cent,  to  be  paid  Jan.  1st,  1851  ? 

30.  What  is  the  interest  on  $1250,75  for  90  days,  at  10 
per  cent? 

31.  What  is  the  amount  of  S71,09  from  Feb.  8th,  1848,  to 
Dec.  7th,  1852,  at  6f  per  cent? 

32.  What  will  be  due  on  a  note  of  $213,27  on  interest 
after  90  days,  at  7  per  cent,  given  May  19th,  1836,  and  pay- 
able October  16th,  1838? 

33.  What  is  the  interest  of  $426,54,  from  Augu<3t  15th, 

1837,  to  March  13th,  1840,  at  7  per  cent  ? 

34.  What   is   the   interest  of  $2132,70,  from  Nov.  17th, 

1838,  to  Feb.  2d,  1839,  at  7i  per  cent? 

35.  What  is  the  interest  of  $38463,  from  April  27th,  1815, 
to  Sept.  2d,  1824,  at  8  per  cent? 

36.  What  is  the  interest  of  $14231,50,  from  June  29th, 
1840,  to  April  30th,  1845,  at  8}  per  cent? 

37.  What  is  the  interest  of  $426,50,  from  Sept.  4th,  1843, 
to  May  4,  1849,  at  9  per  cent? 

38.  What  is  the  interest  of  $4320,  from  Dec.  1st,  1817,  to 
Jan.  22d,  1833,  at  9i  per  cent? 

39.  What  is  the  amount  of  $397,16,  from  March  23,  1824, 
to  March  31st,  1835,  at  101  per  cent  ? 

40.  What  is  the  amount  of  $328,12,  from  July  4th,  1809, 
to  Feb.  15th,  1815,  at  3  per  cent  ? 

41.  What  is  the  amount  of  $164,60,  from  Sept.  27th,  1845, 
to  March  24th,  1855,  at  1^  per  cent? 

42.  What  is  the  amount  of  $1627,50,  from  July  4th,  1839. 
to  August  Ifct,  1855,  at  8  per  cent  ? 


PAKTIAI.    rATMENTS.  247 


CASE    III. 

256.  When  the  principal  is  in  pounds  shillings  and 
pence. 

1.  What  is  the  interest,  at  7  per  cent,  of  £27  I65.  Qri., 
for  2  years  ? 

OPERATION. 

Analysis.— The  interest  on  pounds  £27  155.  9^Z.=^27.7875 
and  decimals  of  a  pound  is  found  in  _07 

the  same  way  as  the  interest  on  doJ-  '^ 

lars  and  decimals  of  a  dollar:  after  l.J4ol20 

which  the  decimal  part  of  the  interest  ^ 

may  be   reduced    to    shillings    and  £3.890250 

pence  :  hence,  £.89025  =z  1 75.  ^d. 

Ans,  £3  175.  ^d. 

1.  Reduce  the  shillings  and  pence  to  the  decimal  of  a 
pound  and  afinex  the  result  to  the  pou7ids. 

II.  Find  the  interest  as  though  the  sum  were  United 
States  Money,  after  which  reduce  the  decimal  part  to  shil- 
ings  and  pence. 

2.  What  is  the  interest  of  £67  195.  6c?.,  at  6  per  cent,  for 
3  years  8  months  16  days  ? 

3.  What  is  the  interest  of  £127  155.  4(i.,  at  6  per  cent, 
for  3  years  and  3  months  '] 

4.  What  is  the  interest  of  £107  I65.  lOc?.,  at  7  per  cent, 
for  3  years  6  months  and  6  days  ? 

5.  What  will  £279  135.  8<i.  amount  to  in  3  years  and  a 
half,  at  51  per  cent  per  annum  ? 

PARTIAL  PAYMENllS. 

257.  A  Partial  Payment  is  a  payment  of  a  part  of  a  note 
or  bond. 

Wc  shall  give  the  rule  established  in  New  York  (see 
Johnson's  Chancery  Jleports,  vol.  i.  page  17),  for  computing 
the  interest  on  a  bond  or  note,  when  partial  payments  have 
i»een  made.  The  same  rule  is  also  adopted  in  Massachusetts, 
and  in  most  of  the  other  states. 

256.  How  do  you  find  the  interest  when  the  principal  is  in  pounds, 
shillings  and  peace  ^ 


248  PARTIAI.    PATIklENTS. 

Rule. — I.  Compute  the  interest  on  the  principal  to  the 
time  of  the  first  payment,  and  if  the  payment  exceed  this 
interest,  add  the  interest  to  the  principal  andfrgm  the  sum 
subtract  the  payment :  the  remainder  forms  a  ntiv  principal: 

II.  But  if  the  payment  is  less  than  the  interest,  take  7io 
fwtice  of  it  until  other  payments  are  made,  which  in  all, 
shall  exceed  the  interest  computed  to  the  time  of  the  last 
payment  :  then  add  the  interest,  so  computed,  to  the  princi- 
pal, and  from  trie  sum  subtract  the  sum  of  the  payments  : 
the  remainder  will  form  a  new  principal  on  which  interest 
is  to  be  computed  as  before. 

Note. — In  computing  interest  on  notes,  observe  that  the  day  on 
which  a  note  is  dated  and  the  day  on  which  it  falls  due,  are  not 
both  reckoned  in  deiermining  the  time,  hut  one  of  them  is  alvmys 
exchdtd.  Thus,  a  note  dated  on  the  first  day  of  May  and  falling 
due  on  the  16th  ol  June,  will  bear  interest  but  one  month  and 
1  5  days. 

EXAMPLES. 


$349,998  Buffalo,  May  \st,  1826. 

1.  For  value  received,  I  promise  to  pay  James  Wilson  or 
order,  three  hundred  and  forty-nine  dollars  ninety-nine  centii 
and  eight  mills,  with  interest  at  6  per  cent. 

James  Bay  well. 

On  this  note  were  endorsed  the  following  payments  : 
Dec.  25th,  1826     Received  $49,998 
July  10th,  1827  "         I  4,998 

Sept.    1st,   1828  "         $15,008 

June  14th,  1829  "         |99,999 

What  was  due  April  15th,  1830? 

Principal  on  int.  from  May  1st,  1826,     -     -     -  $349,998 
Interest  to  Dec.  25th,  1826,  time  of  first  pay- 
ment, 7  months  24  days 13,649-h 

Amount     -     -  $363,647. 


257.  What  is  a  partial  payment  1     What  is   the  rule   for   computing 
intcrtst  wht-n  llicre  art-  partial  payments  1 


Payment  Dec.  25  th,  exceeding  inte^s'Otyi)^"^  ^IB0;99^  T  '! 
Remamder  for  a  new  principal      "^v^?^"     "     ^^^'^'^49 
Interest  of  $313,649  fiom  Dec.   25/Si^«,^J'pr,— 

June  14th,  1829,  2  years  5  months  l^'^iQ£fir^-^-.ia. 47^1 
Amount  -     -     -     .     -     $360,1211 
Payment,  July  10th,  1827,  less  than   )  g  ^  ggg 

interest  then  due ) 

Payment,  Sept.  1st,  1828     -     -     -     -     15,008 

Iheir  sum  less  than  interest  then  due    $20,006 

Payment,  June  14th,  1829  -     -     -     -     99,999 

Their  sum  exceeds  the  interest  then  due      -     -   1^120,005 

Remainder  for  a  new  principal,  June  14,  1829,    $240,1161 

Interest  of  $240,168  fiom  June  14th,  1829,  to 

April  15th,  1830,  10  months  1  day    -     -     -    $   12,0458 

Total  due,  April  15th,  1830     -    .^■2o2,l"6T94- 


$3469,32  New  York,  Feb.  6,  1825. 

2.  For  value  received,  I  promise  to  pay  William  Jenks,  or 
order,  three  thousand  lour  hundred  and  sixty-nine  dollars  aud 
thirty-tM'o  cents,  M'ith  interest  from  date,  at  6  per  cent. 

Bill  Spendthrift. 
On  this  note  were  endorsed  the  following  payments  : 

May  16th,  1828,  received  $  545,76. 

May  16th,  1830,         "       $1276,00. 

Feb.     1st,   1831,         "       $2074,72. 
What  remained  due  Aug.  11th,  18321 

3.  A's  note  of  $635,84  was  dated  September  5, '1817,  on 
which  were  endorsed  the  following  payments,  viz.  :  Nov. 
13th,  1819,  $416,08;  May  10th,  1820,  S152,00  :  what  was 
due  March  1st,  1821,  the  interest  being  6  per  cent? 

LEGAL  INTEREST. 

258.  Legal  Interest  is  the  interest  which  the  law  permits 
a  person  to  receive  for  money  which  he  loans,  and  the  laws 
do  not  favor  the  taking  of  a  higher  rate.  In  most  of  the 
States  the  rate  is  fixed  at  6  per  cent ;  in  New  York,  South 
Carolina  and  Georgia,  it  is  7  ;  and  in  some  of  the  State?  the 
rate  is  iixed  as-  high  as  10  j)er  cent 

<) 


250  PROBLEMS   IN   INTliKEST. 

PROBLEMS    IN    INTEREST. 

259.  Tn  all  questions  of  Interest  there  are  four  things  con- 
sidered, viz.  : 

1st,  The  principal ;  2d,  The  rate  of  interest  ;  3g?  The 
lime  ;  and  4/h,  The  amount  of  interest. 

If  three  of  these  are  known,  the  fourth  can  be  found. 

I.  Knowing  the  principal,  rate,  and  time,  to  iind  the  inter- 
est.     This  case  has  already  been  considered. 

II.  Knowing  the  interest,  time,  and  rate,  to  find  the  prin- 
cipal. 

Cast  the  interest  on  one  dollar  for  the  given,  time,  and  then 
divide  the  given  interest  hy  it — the  quotient  will  be  the  princi- 
pal. 

III.  Knowing  the  interest,  the  principal,  and  the  time,  to 
find  the  rate. 

Cast  the  interest  on  the  principal  for  the  given  time  at  1  per 
cent  and  then  divide  the  given  interest  by  it— the  quotient  will 
be  the  rate  of  interest. 

IV.  Knowing  the  principal,  the  interest,  and  the  rate,  to 
find  the  time. 

Cast  the  interest  on  the  given  principal  at  the  given  rate 
for  1  year  arid  then  divide  the  interest  by  it — the  quotient 
will  be  the  time  in  years  and  decimals  of  a  year. 

EXAMPLES. 

1.  The  interest  of  a  certain  sum  for  4  years,  at  7  per  cent, 
is  S266  :   what  is  the  principal  '\ 

2.  The  interest  of  $3675,  for  3  years,  is  |771,75  :  what  is 
the  rate  1 

3.  The  principal  is  $459,  the  interest  $183,60,  a»d  the 
rate  &  per  cent :  what  is  the  time  % 

4.  The  interest  of  a  certain  sum,  for  3  years,  at  6  per  cent, 
is  $40,50  :   what  is  the  principal "? 

5.  The  principal  is  $918,  the  interest  $269,28,  and  the 
rate  4  per  cent  :  what  is  the  time  ? 

258    Whut  is  legal  interest  I 

'i59.  How  many  things  aje  conuitlered  in  every  question  of  interetf,  \ 
NVhat  are  they  !     What  is  the  rule  fur  each  ^ 


OOMPOUNl)   INTEREST.  251 


COMPOUND  INTEREST. 

260.  Compound  Interest  is  when  the  interest  on  a  princi- 
pal, computed  to  a  given  time,  is  added  to  the  principal,  and 
the  interest  then  computed  on  this  amount,  as  on  a  new 
principal.     Hence, 

Compute  the  interest  to  the  time  at  which  it  becomes  due  ; 
then  add  it  to  the  'principal  and.  compute  the  interest  on  the 
amount  as  on  a  new  p7'incipal :  add  the  interest  agaiii  to 
the  principal  and  compv.te  the  interest  as  before ;  do  the 
same  for  all  the  times  at  which  payments  of  interest  become 
due ;  from  the  last  residt  subtract  the  principal,  and  the 
remainder  will  he  the  compound  interest. 

EXAMPLES. 

1 .  What  will  be  the  compound  interest,  at  7  per  cent,  of 
$37*>0  for  2  years,  the  interest  being  added  yearly  ? 

OPERATION. 

$3750,000  principal  for  1st  year. 

13750  X. 07=      262,500  interest     for  1st  year. 

4012,50^0  principal  for  2d      " 

$40  2,50x.07=  _280^875  interest     for  2d      ** 

4293,375  amount  at  2  years. 
1st  principal  3750,000 
iraount  of  interest  $543,375. 

?,  If  the-  interest  be  computed  annually,  what  will  be  the 
compound  interest  on  $100  for  3  years,  at  6  per  cent? 

3.  What  will  be  the  compound  interest  on  $295,37,  at  6 
per  cent,  for  2  years,  the  interest  being  added  annually  ? 

4.  What  will  be  the  compound  interest,  at  5  per  cent,  of 
^1875,  for  4  years'? 

5.  What  is  the  amount  at  compound  interest  of  $250,  for 
2  years,  at  8  per  cent  % 

6.  What  is  the  compound  interest  of  $939,64,  for  3  years, 
at  7  per  cent  1 

7.  What  will  $125,50  amount  to  in  10  years,  at  4  per  cent 
compound  interest? 

5i(iO.  What  of  compound  interetit  1     How  do  you  compute  it  1 


252 


OOMJ'OUNI)    INTEKEST. 


NoT£. — The  operation  is  rendered  much  shorter  .and  easier,  by 
taking  the  amount  of  1  dollar  for  any  time  and  rate  given  in  the 
following  table,  and  multiplying  it  by  the  given  principal;  the 
product  will  be  the  required  amount,  from  which  subtract  the 
given  principal,  and  the  result  will  be  the  compound  interest  * 

TABLE. 

Which  shows  the  amount  of  $1  or  £1.  compound  interest,  from  I  year 
to  20,  and  at  the  rate  of  3,  4,  5,  6,  and  7  per  cent. 


Years. 

3  per  cent. 

4  per  cent. ,5  per  cent. 

6  per  cent. 

7  percent. 

/ears. 

1 

1 

1.03000 

1.04000 

1.05000 

1.06000 

1.07000 

2 

1.06090 

1.18160 

1.10250 

1.12360 

1.14490 

2 

3 

1.09272 

1.12486 

1.15762 

1.19101 

1.22504 

3 

4 

1.12550 

1.16985  1  1  21550 

1.26247 

1.31079 

4 

5 

1.15927 

1.21065  j  1.27628 

1.33822 

1.40255 

5 

6 

1.19405 

1.20531  I  1.34009 

1.41851 

1.50073 

6 

7 

1.22987 

1.31593 

1.40710 

1.50363 

1.60578 

7 

S 

1.26677 

1.36856 

1.47745 

1.59384 

1.71818 

8 

9 

1.30477 

1.42331 

1.55132 

1.68947 

1.83845 

9 

10 

1.34391 

1.48028 

1.62889 

1.79084 

1.96715 

10 

11 

1.38423 

1.53945 

1.71033 

1.89829 

2.10485 

11 

12 

1.42576 

1.60103 

1.79585 

2.01219 

2.25219 

12 

13 

1.46853 

1.66507 

1.88564 

2.13292 

2.40984 

13 

14 

1.51258 

1.73167 

1.97993 

'  2.26090 

2.57853 

14 

15 

1.55796 

1.80094 

2.07892 

2.39655 

2.75903 

15 

16 

1.60470 

1.87298 

2.18287 

2.54035 

2.95216 

16 

17 

1.65284 

1.94790 

2.29201 

2.69277 

3.15881 

17 

18 

1.70243 

2.02581 

2.40661 

2.85433 

3.37993 

18 

19 

1.75350 

2.10684 

2.52695 

3.02559 

3.61652 

19 

20 

1.80611 

2.19112 

2.65329 

3.20713 

3.86968 

20 

Note. — When  there  are  months  and  days  in  the  time,  find  the 
amount  for  the  years,  and  on  this  amount  cast  the  interest  for  the 
months  and  days :  this,  added  to  the  last  amount,  will  be  the  re- 
quired amount  for  the  whole  time. 

8.  What  is  the  amount  of  $96,50  for  8  years  and  6  raoiiths,, 
interest  being  compounded  annually  at  7  per  cent  ? 

9.  What  is  the  compound  interest  of  ^300  for  5  years 
£  months  and  15  days,  at  6  per  cent  ? 

10.  What  is  the  compound  interest  of  $1250  lor  3  years 
3  months  and  24  days,  at  7  per  cent  1 

11.  What  will  $56,50  amount  to  in  20  years  and  4  ruonthjj, 
ut  5  per  cent  compound  interest  l 

*  Tlio  n;>ti't  may  dill'tr  in  tiic  mills  jilatc  (nnii  thai  (»l»tui"cd  l-v  t'i» 
other  rule 


DIBOOUNT.  253 


DISCOUNT. 

261.  Discount  is  an  allowance  made  lor  the  payment  of 
money  before  it  is  due. 

The  face  of  a  note  is  the  amount  named  in  the  note.* 

Note. — Days  of  grace  are  days  allowed  for  the  payment  ol 
a  nole  after  the  expiration  of  the  time  named  on  its  face.  By 
mercantile  usage  a  nole  does  not  legally  fall  due  until  3  days 
after  the  expiration  of  the  time  named  on  its  face,  unless  the  note 
specifies  without  grace. 

Days  of  grace,  however,  are  generally  confined  to  mercantile 
paper  and  to  notes  discounted  at  banks. 

262.  The  present  value  of  a  note  is  such  a  sum  as  being 
put  at  interest  until  the  note  becomes  due,  would  increase  to 
an  amount  equal  to  the  face  of  the  note. 

The  discount  on  a  note  is  the  difference  between  the  face 
of  the  note  and  its  present  value. 

1.  I  give  my  note  to  Mr.  Wilson  for  $x07,  payable  in 
1  year  :  what  is  the  present  value  of  the  note,  if  the  uiterest 
is  7  per  cent  ?  what  the  discount  ? 

ePERATlON. 

Analysis.— Since  1  dollar  in  1  year'  $107-4-1,07  =  $100. 

at  7  per  cent,  will  amount  to  $1,07,  the  proof. 

present   value  will  be  as  many  dollars  Int.  $100  lyr.=^$     7 

as  $1,07  is  contained  times  in  the  face  Principal,                 100 

of  the  note  :  viz.,   $100:  and  the  dis-        .  ^  STTTT^ 
.  „.: n  t. .  tfi. , rv'n     lI^,r^/._<B>rT  .  V Amount,  ^107 

Discount,  7 


count  will  be  $107— $100^$7  :  hence, 


Divide  the  face  of  the  note  by  1  dollar  plus  the  interest  oj 
1  dollar  for  the  yiven  tirne^  and  the  quotient  will  be  tlie  pre- 
sent value:  take  this  sum  from  the  face  of  the  note  and  tlu 
reTiiainder  will  be  the  discount. 


261.  What  is  discount  1    What  is  the  face  of  a  note''    What  are  ilays 
of  grace  1 

262.  What  is  present  value  ?     What  is  the  discount  1     How  do  yuv 
fuiJ  the  present  value  of  a  note  ? 

*  Sec  Appendix,  page  UJO. 


254  DISCOUNT. 

EXAMPLES. 

1.  What  is  the  present  value  of  a  note  for  $1828,75,  due 
in  1  year,  and  bearing  an  interest  of  41  per  cent  ? 

2.  A  note  of  $1651,50  is  due  in  11  months,  but  the  person 
to  whom  it  is  payable  sells  it  with  the  discount  off  at  6  per 
cent  :  how  much  shall  he  receive  ? 

Note. — When  payments  are  to  be  made  at  different  times,  find 
the  present  value  of  the  sums  separately,  and  their  sum  will  be  the 
present  value  of  the  note. 

3.  What  is  the  present  value  of  a  note  for  $10500,  on  which 
1900  are  to  be  paid  in  6  months  ;  $2700  in  one  year  ;  $3900 
in  eighteen  months  /  and  the  residue  at  the  expiration  of  two 
years,  the  rate  of  interest  being  6  per  cent  per  annum  ? 

4.  What  is  the  discount  of  £4500,  one-half  payable  in  six 
months  and  the  other  half  at  the  expiration  of  a  year,  at  7 
per  cent  per  annum  1 

5.  What  is  the  present  value  of  $5760,  one-half  payable  in 

3  months,  one-third   in  6  months,  and  the  rest  in  9  months, 
at  6  per  cent  per  annum  ? 

6.  Mr.  A  gives  his  note  to  B  for  $720,  one-half  payable  m 

4  months  and  the  other  half  in  8  months  ;  what  is  the  present 
value  of  said  note,  discount  at  5  per  cent  per  annum  ? 

7.  What  is  the  diiierence  between  the  interest  and  discount 
of  $750,  due  nine  months  hence,  at  7  per  cent? 

8.  What  is  the  present  value  of  $4000  payable  in  9  months, 
discount  4J  per  cent  per  annum  ? 

9.  Mr.  Johnson  has  a  note  against  Mr.  Williams  for 
$2146,50,  dated  August  17th,  1838,  which  becomes  due  Jan. 
11th,  1839  :  if  the  note  is  discounted  at  6  percent,  what 
ready  money  must  be  paid  for  it  September  25th,  1838  1 

10.  C  owes  D  $3456,  to  be  paid  October  27th,  1842;  C 
wishes  to  pay  on  the  24th  of  August,  1838,  to  which  D  con- 
sents ;  how  much  ought  D  to  receive,  interest  at  6  per  cent  ? 

11.  What  is  the  present  value  of  a  note  of  $4800,  due  4 
years  hence,  the  interest  being  computed  at  5  per  cent  per 
annum  ? 

12.  A  man  having  a  horse  for  sale,  offered  it  for  $225  cash 
in  hand,  or  $230  at  9  months ;  the  buyer  chose  the  latter  : 
did  tlie  seller  lose  or  make  by  his  offer,  supposing  money  to 
be  worth  7  per  cent  ? 


BANK    DISUUUNT.  256 


BANK  DISCOUNT. 

263.  Bank  Discount  is  the  charge  made  by  a  bank  for  the 
payment  of  money  on  a  note  before  it  becomes  due. 

By  the  custom  of  banks,  this  discount  is  the  interest  on  the 
amount  named  in  a  note,  calculated  from  the  time  the  note 
is  discounted  to  the  time  when  it  falls  due  ;  in  which  time 
the  three  days  of  grace  are  always  included. 

The  interest  is  always  paid  in  advance. 

Rule. — Add  3  da^JS  to  the  time  which  the  note  has  to  run, 
and  then  calculate  the  interest  for  that  time  at  the  given  rate, 

EXAMPLES. 

1.  What  is  the  bank  discount  of  a  note  for  $350,  payable 

0  months  after  date,  at  7  per  cent  interest  ? 

2.  What  is  the  bank  discount  of  a  note  of  $1000  payable 
in  60  days,  at  6  per  cent  interest  ? 

3.  A  merchant  sold  a  cargo  of  cotton  for  $15720,  for  which 
he  receives  a  note  at  6  months  :  how  much  money  will  he 
receive  at  a  bank  for  this  note,  discounting  it  at  6  per  cent 
interest  ? 

4.  What  is  the  bank  discount  on  a  note  of  $556,27  paya- 
ble in  60  days,  discounted  at  6  per  cent  interest  ? 

5.  A  has  a  note  against  B  for  $3456,  pa)  able  in  three 
months  ;  he  gets  it  discounted  at  7  per  cent  interest .  how 
much  does  he  receive  % 

&^  What  is  the  bank  discount  on  a  note  of  $367.47,  having 

1  year,  1  month,  and  13  days  to  run,  as  shown  by  the  face  of 
the  note,  discounted  at  7  per  cent  1 

7.  For  value  received,  I  promise  to  pay  to  John  Jones,  on 
the  20th  of  November  next,  six  thousand  five  hundred  and 
seventy-nine  dollars  and  15  cents.  What  will  be  the  discount 
on  this,  if  discounted  on  the  1st  of  August,  at  6  per  cent  per 
annum  1 

263.  What  is  bank  discount  ?  How  is  interest  calculated  by  the 
custom  of  banks  ^  How  is  the  intereftt  paid  1  How  do  you  liuJ  U»« 
mteicKtl 


'256  liANX    DIBOOUNT. 

8.  A  merchant  bought  175  barrels  of  flour  at  $7,50  cents 
a  barrel,  and  sells  it  immediately  for  $9,75  a  barrel,  for 
which  he  receives  a  good  note,  payable  in  6  months.  If  he 
should  get  this  note  discounted  at  a  bank,  at  6  per  cent,  what 
will  be  his  gain  on  the  flour '? 

264.  To  make  a  note  due  at  a  future  iime,  whose  present 
value  shall  be  a  given  amount. 

1.  For  what  sum  must  a  note  be  drawn  at  3  months,  so 
that  when  discounted  at  a  bank,  at  6  per  cent,  the  amount 
received  shall  be  $500  1 

Analysis. — If  we  find  the  interest  oii  1  dollar  for  the  given 
time,  and  then  subtract  that  interef^t  from  1  dollar,  the  remainder 
will  be  the  present  value  of  1  dollar,  due  at  the  expiration  of  that 
time.  Then,  the  number  of  times  which  the  present  value  of 
the  note  contains  the  present  value  of  1  dollar,  will  be  the  num- 
ber of  dollars  for  which  the  note  must  be  drawn  :  hence, 

Divide  the  present  value  of  the  note  by  the  present  value  of 
1  dollar,  reckoned-  for  the  saw.e  time  and  at  the  same  rate  of 
interest^  and  the  quotient  will  be  the  face  of  the  note. 

OPERATION. 

Interest  ol  $1  for  the  time,  3mo.  and  3</a.=$0.0155,  which 
taken  from  $1,  gives  present  value  of  $1=0,9845  ;  then,  $500-^- 

0,9845  =  $507,872  +  =face  of  note. 

PROOF. 

Bank  interest  on  $507,872  for  3  months,  mcluding  3  days  of 
grace,  at  6  per  cent=7,872,  which  being  taken  from  the  face  of 
the  note,  leaves  $500  for  its  present  value. 

EXAMPLES. 

1.  For  what  sum  must  a  note  be  drawn,  at  7  per  cent, 
payable  on  its  face  in  1  year  6  months  and  15  days,  so  that 
when  discounted  at  bank  it  shall  produce  $307,27  1 

2.  A  note  is  to  be  drawn  having  on  its  face  8  months  and 
12  days  to  run,  and  to  bear  an  interest  of  7  per  cent,  so  that 
it  will  pay  a  debt  of  $5450  :  what  is  the  amount  ? 

204.  How  do  you  make  a  note  payable  at  a  future  time,  whose  pro- 
:itnt  v;ilue  .shall  he  a  y^ivfii  aiii'iuut  1 


K(ii-ATiON  OF  payml:nts.  1557 

3.  What  sum,  6  months  and  9  days  from  July  18th,  1856, 
diawing  an  interest  of  6  per  cent,  will  pay  a  debt  of  $674,89 
at  bank,  on  the  Ist  of  August,  1856  ? 

4.  Mr.  Johnson  has  Mr.  Squires'  note  for  $874,^)7,  having 
4  months  to  run,  from  July  13th,  without  interest.  On  the 
first  of  October  he  wishes  to  pay  a  debt  at  bank  of  $750, 25, 
and  discounts  the  note  at  5  per  cent  in  payment  :  how  much 
must  he  receive  back  from  the  bajik  ? 

5.  Mr.  Jones,  on  the  1st  of  June,  desires  to  pay  a  debt  at 
bank  by  a  note  dated  May  16lh,  having  6  monilis  to  run  and 
draM'ing  7  per  cent  interest  :  for  what  amount  must  the  note 
be  drawn,  the  debt  being  ^1683,75  ? 

6.  Mr.  Wilson  is  indebted  at  the  bank  in  the  sum  oi 
$367,464,  which  he  wishes  to  pay  by  a  note  at  4  months 
■with  interest  at  7  per  cent :  lor  what  amount  must  the  note 
be  drawn  ] 

EQUATION  OF  PAYMENTS. 

265.  Equation  of  Payments  is  the  operation  of  finding  the 
mean  time  of  payment  of  several  sums  due  at  difi'erent  times, 
BO  that  no  interest  shall  be  lost  or  gained.* 

1.  If  I  OM^e  Mr.  Wilson  2  dollars  to  be  paid  in  6  months, 
3  dollars  to  be  paid  in  6  months,  and  1  dollar  to  be  paid  in 
12  months,  what  is  the  mean  time  of  payment  ? 

OPERATION. 

Int.  of  $2  for    6wo.  =  int.  of  $1  for  l2?no.  2x    6:ir:12 

"     of  $3  for     8mo.  =  int.  of  $1  for  24mo.  3x    8zz:24 

"     of  $1   for  12mo.  =  int.  of  SI  for  \2mo.  1  X  12=_]^ 

$6                                                 48  48 

Analysis. — The  interest  on  all  the  sums,  to  the  times  of  pay^ 
ment,  is  equal  to  the  interest  of  $1  for  48  months.  But  48  is 
equal  to  the  sum  of  all  the  products  which  arise  from  muliiplying 
each  sum  by  the  time  at  which  it  becomes  due  :  hence,  the  sum 
of  the  products  is  equal  to  the  time  which  would  be  necessary  for 
$1  to  produce  the  same  interest  as  would  be  produced  by  all  the 
principals. 

*  The  mean  time  of  payment  is  sometimes  found  by  first  finding  the 
present  value  of  each  payment ;  but  the  rule  here  given  has  the  sane* 
tioii  of  the  best  authorities  ui  this  country  and  England. 

17 


258  EQUATION    OJ'^    PAYMENTS. 

If  Si  wiL  produce  a  certain  interest  in  48  months,  in  what  time 
will  $6  (or  the  sum  of  the  payments)  produce  the  same  interest  ? 
The  time  is  obviously  found  by  dividing  48  (the  sum  of  the  pro- 
ducts) by  $6,  (the  sum  of  the  payments.) 

Hence,  to  find  the  mean  time, 

Multiply  each  payment  hy  the  time  before  it  becomes  due^ 
and  divide  the  sum  of  the  jyroducts  by  the  sum  of  the  pay- 
ments :  the  quotient  will  be  the  mean  time. 

EXAMPLES. 

1.  B  owes  A  $600  ;  ^200  is  to  be  paid  in  two  months, 
$200  in  four  months,  and  $200  in  six  months  :  what  is  the 
mean  time  for  the  payment  of"  the  whole  ? 

OPERATION. 

Analysis.— We   here   multiply  each  ^^^^^^   ^^^ 

Bum   by  the  time  at  which   it   becomes  200x4—    800 

due,  and  divide  the  sum  of  the  products  200  X  6=  1200 

by  the  *um  of  the  payments.  5|00        )24|00 

Ans.  4    months. 

2.  A  merchant  owes  |600,  of  which  ^100  is  to  be  paid  in 
4  months,  $200  in  10  months,  and  the  remainder  in  16 
months  :  if  he  pays  the  whole  at  once,  in  what  time  must  he 
make  the  payment  ? 

3.  A  merchant  owes  |600  to  be  paid  in  12  months,  8800 
to  be  paid  in  6  months,  and  $900  to  be  paid  in  9  months  : 
what  is  the  equated  time  of  payment  ? 

4  A  owes  B  $600  ;  one-third  is  to  be  paid  in  6  months, 
one- fourth  in  8  months,  and  the  remainder  in  12  months  : 
what  is  the  mean  time  of  payment  ? 

5.  A  merchant  has  due  him  $300  to  be  paid  in  60  days, 
1500  to  be  paid  in  120  days,  and  S750  to  be  paid  in  180 
days :  what  is  the  equated  time  for  the  payment  of  the 
whole  1 

6.  A  merchant  has  due  him  $1500  :  one-sixth  is  to  be 
paid  in  2  months,  one-third  in  3  months,  and  the  rest  in  6 
months  :  what  is  the  equated  time  for  the  payment  of  the 
whole  ? 

2G.').  What  is  equatiuti  of  payments  1  How  do  you  find  th  mean  or 
equated  time  1 


htiUATION    OF    I'AYMLI^TS.  259 

7  I  owe  $1000  to  be  paid  on  the  first  of  January,  $1500 
on  the  1st  of  February,  $3000  on  the  1st  of  March,  and 
$4000  on  the  15th  of  April  :  reckoning  from  the  1st  of  Janu- 
ary, and  calling  February  28  days,  on  what  day  must  the 
money  be  paid  ? 

Note. — If  one  of  the  payments,  as  in  the  above  example,  is  due 
on  the  day  from  which  the  equated  time  is  reckoned,  its  corres- 
ponding product  will  be  nothmg,  but  the  payment  must  still  be 
added  in  finding  the  sum  of  the  payments. 

8.  i  owe  Mr.  Wilson  $100  to  be  paid  on  the  15th  of  July, 
$200  on  the  15th  of  August,  and  300  on  the  9th  of  Septem- 
ber :  what  is  the  mean  time  of  payment  1 

OPERATION. 

From  1st  of  July  to  1st  payment  14  days. 
"        "  "      to  2d  payment  45  days. 

'*        "  "      to  3d  payment  70  days. 

100X14—    1400 

200x45:-r   9000 

Then  by  rule  given  above  we      300  X  70  =  21000 

*^^^®'  600     6|00j314|Q0 

~52i 

Hence,  the  equated  time  is  52i  days  from  the  1st  of  July ;  that 
is,  on  the  22d  day  of  August. 

But  if  we  estimate  the  time  from  the  15th  of  July  we  shall  have 

From  July  15th  to  1st  payment     0  days. 

"  '*  to  2d  payment  30  days. 

^  **  "  to  3d  payment  54  days. 

Then,  100  x    0=       000 

200x30==     6000 
300x54--^  16200 

"600     6|00)222iQQ" 
~37~ 
Hence,  the  payment  is  due  in  37  days  from  July  15tl'  .  cr,  on 
the  22d  of  August — the  same  as  before. 

Therefore :  Any  day  may  be  taken  as  the  one  from  luhich 
the  mean  time  is  reckoned. 

Note. — If  one  payment  is  due  on  the  day  from  which  the  time  is 
reckoned,  how  do  you  treat  iti  Can  you  compute  the  time  from  any 
day  ? 


200  AfeSKSSlISJG    TAXKS. 

9.  Mr.  Jones  purchased  of  Mr.  Wilson,  on  a  credit  of  six 
mouths,  goods  to  the  following  amounts  : 

15th  of  January,     a  bill  of  |3750, 

10th  of  February,  a  bill  of    3000, 

6th  of  March,        a  bill  of    2400, 

8th  of  June,  a  bill  of    2250. 

He  wishes,  on  the  1st  of  July,  to  give  his  note  for  the 
amount  :  at  what  time  must  it  be  made  payable  ? 

10  Mr.  Gilbert  bought  $4000  worth  of  goods  :  he  was  to 
pay  $1600  in  five  months,  $;1200  in  six  months,  and  the  re- 
mainder in  eight  months  :  what  will  be  the  time  of  credit,  if 
he  pays  the  whole  amount  at  a  single  payment  1 

11.  A  merchant  bought  several  lots  of  goods,  as  follows  : 
A   bill  of  $650,  June      6th, 
A  bill  of    890,  July       8th, 
A  bill  of  7940,  August  Ist. 

Now,  if  the  credit  is  6  months,  how  many  days  from  De- 
cember 6th  before  the  note  becomes  due]     At  what  time  1 

ASSESSING    TAXES. 

266.  A  tax  is  a  certain  sum  required  to  be  paid  by  the 
inhabitants  of  a  town,  county,  or  state,  for  the  support  of 
government.  It  is  generally  collected  from  each  individual^ 
in  proportion  to  the  amount  of  his  property. 

In  some  states,  however,  every  white  male  citizen  over  the 
age  of  twenty-one  years  is  required  to  pay  a  certain  tax. 
This  tax  is  called  a  poll-tax  ;  and  each  person  so  taxed  is 
called  a  jjoll. 

267.  In  assessing  taxes,  the  first  thing  to  be  done  is  to  make 
a  complete  inventory  of  all  the  property  in  the  town  on  which 
the  tax  is  to  be  laid.  If  there  is  a  poll-lax,  make  a  full  list 
of  the  polls  and  multiply  the  number  by  the  tax  on  each 
poll,   and   subtract   the   product  from  the  whole  tax   to   be 

26G.  What  is  a  tax  1  How  is  it  generally  collected  ?  What  it;  a 
poll-taxi 


\BSKSSrNG    TAXICS.  2(51 

raised  by  the  to>\Ti  :  the  remainder  will  be  the  amount  to 
be  raised  on  the  properly.  Having  done  this,  divide  the 
whole  tax  to  be  raised  by  the  amount  of  taxable  'pmpeny, 
and  the  quotient  will  be  tJte  tax  on  ^1.  Then  multiply  thie 
quotient  by  the  inventory  of  each  individual,  and  the  product 
will  be  the  tax  on  his  property. 

EXAMPLES. 

1.  A  certain  town  is  to  be  taxed  $4280  ;  the  property  on 
which  the  tax  is  to  be  levied  is  valued  at  $1000000.  Now 
there  are  200  polls,  each  taxed  $1,40.  The  property  of  A 
is  valued  at  S2800,  and  he  pays  4  polls 

B's  at  $2400,  pays  4  polls.     E's  at  $7242,  pays  4  polls. 
C's  at  $2530,  pays  2     "  F's  at  $1651,  pays  6     " 

D's  at  $2250,  pays  6     "         G's  at  $1600,80  pays  4  " 

What  will  be  the  tax  on  1  dollar,  and  what  will  be  A's 
tax,  and  also  that  of  each  on  the  list  ? 

First,       $1,40x200  =  $280  amount  of  poll-tax. 
$4280— $280  =  4000  amount  to  be  levied  on  proi.»erty. 
Then,      $4000-h$1000000  =  4  mills  on  $1. 
Now,  to  find  the  tax  of  each,  as  As,  for  example, 


A's  inventory     -     -     - 

$2800 
,004 

11,20 

4  polls  at  $1,40  each 

5,60 

A's  whole  tax  -     -     - 

$16,80 

In  the  same  manner  the  tax  of  each  person  in  the  town- 
ship may  be  found. 

Having  found  the  per  cent,  or  the  amount  to  be  raised  on 
each  dollar,  form  a  table  showing  the  amount  which  certain 
sums  would  produce  at  the  same  rate,  per  cent.  Thus,  after 
having  found,  as  in  the  last  example,  that  4  mills  are  to  be 
raised  on  every  dollar,  we  can,  by  multiplying  in  succession 
by  the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  &c.,  form  the  following 

267.  WTiat  is  the  first  thing  to  be  done  in  assessing  a  tax  1  If  there 
IS  a  poll-tax,  how  do  you  find  the  amount]  How  then  do  you  find  the 
per  cent  of  tax  to  be  levied  on  a  dollar  1  How  do  you  then  find  the 
ajjjount  to  be  levied  on  cat-h  individual  t 


202 


ASSESSING    TAXES. 


TABLE. 


$ 

$ 

$ 

$ 

$      $ 

1  gives  0.004 

20  gives  0.080 

300  gives  1.200 

2   ' 

'  0.008 

30 

'  0.120 

400  "   1.600 

•  3   ' 

*  0.012 

40 

'  0.160 

500  "   2.000 

4   ' 

♦  0.016 

50 

'  0.200 

600  "   2.400 

5   ' 

'  0.020 

60 

«  0.240 

700  •'   2.800 

6   ' 

'  0.024 

70 

'  0.280 

800  "   3.200 

7   ' 

'  0.028 

80 

'  0.320 

900  "   3.600 

8   ' 

'  0.032 

90 

'  0.360 

1000  "   4  000 

9   ' 

'  0.036 

100 

'  0.400 

2000  "   8.000 

10   * 

'  0.040 

200 

'  0.800 

3000  "  12.000 

This  table  shows  the  amount  to  be  raised  on  each  sum  in 
the  columns  under  $'s. 

EXAMPLES. 

1.  Find  the  amount  of  B's  tax  from  this  table. 

B's  tax  on  12000     -     -     is     -     $8,000 

B's  tax  on      400     -     -     is     -     $1,600 

B's  tax  on  4  polls,  at  $1,40     -     $5,600 

B's  total  tax     -     -     is     -  $15,200 

2.  Find  the  amount  of  C's  tax  from  the  table. 


C's  tax  on  $2000 
C's  tax  on      5a0 
C's  tax  on        30 
C's  tax  on  2  polls 
C's  total  tax     - 


-  $8,000 

-  $2,000 

-  $0,120 

-  $2,800 
s     -  $12,920 


paid 


In  a  similar  manner,  we  might  find  the  taxes  to  be 
by  D,  E,  &c. 

3.  If  the  people  of  a  town  vote  to  tax  themselves  $1500, 
to  build  a  public  hall,  and  the  property  of  the  town  is  valued 
at  $300,000,  what  is  D's  tax,  whose  property  is  valued  at 
$2450? 

4.  In  a  school  district  a  school  is  supported  by  a  tax  on 
the  property  of  the  district  valued  at  $121340.  A  teacher  is 
employed  lor  5  months  at  $40  a  month,  and  contingent  ex- 
penses are  $42,68  ;  what  will  be  a  farmer's  tax  whose  property 
is  valued  at  $3125  ? 


CX)TNS    AND   GUKEENOY.  268 


COINS    AND    CURRENCY. 

268.  Coins  are  pieces  of  metal,  of  gold,  silver,  or  copper,  of 
fixed  values,  and  impressed  with  a  public  stamp  prescribed 
by  the  country  where  they  are  made.  These  are  called 
specie,  and  are  declared  to  be  a  legal  tender  in  payment  of 
debts.  The  Constitution  of  the  United  States  provides,  that 
gold  and  silver  only  shall  be  a  legal  tender. 

269.  Currency  is  what  passes  for  money.  In  our  country 
there  are  three  kinds. 

1st.  The  coins  of  the  country  : 

26?.  Foreign  coins,  having  a  fixed  value  estabhshed  by 
law  : 

3d.  Bank  notes,  redeemable  in  specie. 

Note. — Tfie  foreign  coins  most  in  use  in  this  country  are  the 
English  shilling,  valued  at  22  cents  2  mills  ;  the  English  sove- 
reign, valued  at  $4,84  ;  the  French  franc,  valued  at  18  cents  6 
mills  ;  and  the  five-franc  piece,  valued  at  $0,93. 

Although  the  currency  of  the  United  States  is  in  dollars, 
cents  and  mills,  yet  in  some  of  the  States  accounts  are  still 
kept  in  pounds,  shillings  and  pence. 

In  all  the  States  the  shilling  is  reckoned  at  12  pence,  the 
pound  at  20  shillings,  and  the  dollar  at  100  cents. 

The  following  table  shows  the  number  of  shillings  in  a  dol- 
lar, the  value  of  £1  in  dollars,  and  the  value  of  $1.  in  the 
fraction  of  a  pound  : 


In  English  currency, 

In  N.  E.,  Va.,  Ky.,  ) 

Tenn.,                    J 

4s.  6d. 
16s. 

-  £l  =  $4,84,  and  $l  =  £^l^. 
.  £1  =  $3^,     andll^XyV 

In  N.  Y.,  Ohio,  N. 

Carolina, 
In  N.  J.,  Pa.,  Del.  ) 

Md., 

8s. 
7s.  6d. 

-  £1-$2|, 

-  £1=$2|, 

and  $1=£  |. 
and  $!  =  £    f. 

In  S.  Carolina  &  Ga. 

In  Canada  &  Nova  ) 

Scotia,                     J 

4s.  8d. 
5s. 

-£l  =  $4f, 
-£1=S4, 

and$l=£  ^. 
and8l=£    i 

268.  What  are  coins- 
legal  tender  ? 

What  are  they  called  1 

What  is  iiiatie  a 

204  KI'ID'.C'J ION    OF    CUKICKNOIES. 


REDUCTIOxN  OF  CURRENCIES. 

270.  Reduction  of  Currencies  is  changing  their  denoniiiia 
tions  without  changing  their  values. 

There  are  two  cases  of  the  Reduction  of  Currencies  : 

1st.  To  change  a  currency  in  pounds  shillings   and  pence, 

to  United  States  currency. 

2d.    To  change  United  States  currency  to  pounds,  shillings 

and  pence. 

271.  To  reduce  pounds,  shillings  and  pence  to  United 
States  currency. 

1.  What  is  the  value  of  £3  126".  6c/.,  New  England  cur- 
rency, in  United  States  money  ? 

OPERATION. 

Analysis.— Since   £l=$3i,   the      £3   125.   6rf.  =  £3.625 
number  of  dollars  in  £3  125.  6d.=       (JqHs   in  £1    =  3i 

£3.625,  will    be   equal    to  £3.625  "  To^l_ 

taken  3i  times  :  that  is,  to  $12,08  :  l.^U»-h 

hence,  10.875 

Ans.  $12.0S3-f- 

Multiply  the  amount  reduced  to  pounds  and  the  decimals  of 
a  pound  by  the  number  of  dollars  in  a  jjound^  and  the  product 
will  be  the  answer. 

272.  To  reduce  United  States  money  to  pounds,  shillings 
and  pence. 

1.  What  is  the  value  of  |375,87,  in  pounds,  shillings  and 
pence,  New  York  currency  ] 

Analysis. —  Since    $1  =£f;    the 
number  of  pounds  in  $375.87  will  be  operation. 

equal  to  this  number  taken  f  times  :       $375.87  xf  =  £150.348 
thatis,  equal  to  £150.348 =£150  65.  =£150   6s.  \l^d. 

ll^d.  :  hence, 

269.  What  is  currency  ■?  How  many  kiuds  are  there  1  "WTiat  foreijm 
coins  are  most  used  in  this  country  !  What  are  the  denominations  of 
United  States  currency  ]  What  denominations  are  sometimes  used  in 
the  States  ? 

270.  What  is  reduction  of  currencies  ?  How  many  kinds  of  reduc- 
tion are  there  1     What  are  they  ! 

271.  What  is  the  rule  for  reducing  from  pounds,  slulliug.s  and  pence 
to  United  States  uiouey  ] 


EXCHANGE.  205 

Multiply  the  amount  by  that  fraction  of  a  pound  which 
denotes  the  value  of$'[,  and  the  product  will  be  the  anawei 
in  pounds  and  decimals  of  a  pound. 

EXAMPLES. 

1.  What  is  the  value  of  £127  l&s.  6c?.,  New  England 
currency,  in  United  States  moiiey  1 

2.  What  is  the  value  of  ^2663.75  in  pounds,  shillings  and 
pence,  Pennsylvania  currency  ? 

3.  What  is  the  value  of  X'4&9  Ss.  6rf.,  Georgia  currency,  in 
United  States  money  ? 

4.  What  is  the  value  of  $973,28  in  pounds,  shillings  and 
pence,  North  Carolina  currency? 

5.  W^hat  is  the  value  in  United  States  money  of  £637  18«. 
8cZ.,  Canada  currency] 

6.  Reduce  ^102,85  to  English  money  ;  to  Canada  cur- 
rency ;  to  New  England  currency  ;  to  New  York  cunency  ; 
to  Pennsylvania  currency  ;  to  South  Carolina  currency. 

7.  Reduce  £51  136-.  0\d.  English  money;  £62  lO*-.  Can- 
ada currency;  £75  New  England  currency;  £100  New 
York  currency  ;  £193  15.v.  Pennsylvania  currency;  and  £58 
66.  l\d.  Georgia  currency,  to  United  States  money. 

EXCHANGE. 

273.  Exchange  denotes  the  payment  of  a  sum  of  money 
by  a  person  residing  in  one  place  to  a  person  residing  in  an- 
other. The  payment  is  usually  made  by  means  of  a  bill  of 
exchange. 

A  Bill  of  Exchange  is  an  order  liom  one  person  to  anothei 
directing  the  payment  to  a  third  person  named  therein  of  a 
certain  sum  of  money  : 

1.  He  who  writes  the  open  letter  of  request  is  called  tne 
drawer  or  maker  of  the  bill. 

2.  The  person  to  whom  it  is  directed  is  called  the  drawee. 

272  What  is  the  rule  for  reducing  from  United  States  money  to 
pounds,  shillings  and  pence  ? 

273.  What  does  exchange  denote  \  How  is  the  payment  generally 
made  ]  What  is  a  bill  of  exchange  ]  Who  is  the  drawer  1  Who  the 
drawee  1     Who  the  buver  or  remitter  ^ 


260  PORMGN   BILLS. 

3.  The  person  to  whom  the  money  is  ordered  to  be  paid  is 
called  the  'payee  ;  and 

4.  Any  person  who  purchases  a  bill  of  exchange  i&  called 
the  buyer  or  remitter. 

274.  A  bill  of  exchange  is  called  an  inland  bill,  when  the 
drawer  and  drawee  both  reside  in  the  same  country  ;  and  when 
they  reside  in  different  countries,  it  is  called  a  foreign  bill. 

Exchange  is  said  to  be  at  par,  when  an  amount  at  the 
place  from  which  it  is  remitted  will  pay  an  equal  amount  at 
the  place  to  which  it  is  remitted.  Exchange  is  said  to  be  at 
a  premium,  or  above  par,  when  the  sum  to  be  remitted  will 
pay  less  at  the  place  to  which  it  is  remitted  ;  and  at  a  dis- 
count^ or  below  par,  when  it  will  pay  more. 

EXAMPLES. 

1.  A  merchant  at  Chicago  wishes  to  pay  a  bill  in  New 
York  amounting  to  $3675,  and  finds  that  exchange  is  1^  per 
cent  premium  :  what  must  he  pay  for  his  bill  ? 

2.  A  merchant  in  Philadelphia  wishes  to  remit  to  Charles- 
ton $8756,50,  and  finds  exchange  to  be  1  per  cent  below  par  ; 
what  must  he  pay  for  the  bill  ? 

3.  A  merchant  in  Mobile  wishes  to  pay  in  New  York 
$6584,  and  exchange  is  2^  per  cent  premium  :  how  much 
must  he  pay  for  such  a  bill  ? 

4.  A  merchant  in  Boston  wishes  to  pay  in  New  Orleans 
$4653,75  ;  exchange  between  Boston  and  New  Orleans  is  1-^ 
per  -cent  below  par  :  what  must  he  pay  for  a  bill  1 

5.  A  merchant  in  New  York  has  $3690  which  he  wishes 
to  remit  to  Cincinnati  ;  the  exchange  is  Ij  per  cent  belo"w 
par  :  what  will  be  the  amount  of  his  bill  1 

FOREIGN  BILLS. 

275.  A  Foreign  Bill  of  Exchange  is  one  in  which  the 
drawer  and  drawee  live  in  different  countries. 

Note. — In  all  Bills  of  Exchange  on  England,  the  £  sterling  is 
the  unit  or  base,  and  is  still  reckoned  at  its  former  value  of  $4| 
=$4,4444+,  instead  of  its  present  value  $4,84. 

274.  When  is  a  bill  of  exchange  said  to  be  inland  1  When  foreign  1 
When  is  exchange  said  to  be  at  pari  When  at  a  premium  ''  When 
at  a  discount  ^ 


FOREIGN    HILLS.  267 

Hence,  £1=^4.4444-1- 

Add  9  per  cent,  -3999 

Gives  the  present  value  of  £l  $4.8443. 

Hence,  the  true  'par  value  of  Exchange  on  England  is 
9  per  cent  on  the  nominal  base. 

1.  A  merchant  in  New  York  wishes  to  remit  to  England 
a  bill  of  Exchange  for  £125  155.  6<^ :  how  much  must  he 
pay  for  this  bill  when  exchange  is  at  9^  per  cent  premium  ? 

-     £125  155.  6^/. =£125.775 

Add  91  per  cent      -     -     -     -  11.9486  + 

gives  amount  in  £'s,  at  $4|  =  ^.  £137.7236  + 
Note. — The  pounds  and  decimals  of  a  pound   are  reduced  to 
dollars  by  multiplying  by  40   and  dividing  by  9 — giving,  in  this 
case,  $612,105. 

Rule. — 1.   Reduce  the  amount  of  the  bill  to  pounds  and 

decimals  of  a  pound,  and  then  add  the  preTniumof  exc]ia7ige. 

II.   Multiply  the  result  by  40  and  divide  the  product  by 

9  :  the  quotient  will  be  the  answer  in  United  States  Money. 

2.  A  merchant  shipped  100  bales  of  cotton  to  Liverpool, 
each  weighing  450  pounds.  They  were  sold  at  l\d.  per 
pound,  and  the  freight  and  charges  amounted  to  £187  IO5. 
He  sold  his  bill  of  exchange  at  9|-  per  cent  premium  :  how 
much  should  he  receive  in  United  States  Money  ? 

3.  There  were  shipped  from  Norfolk,  Va.,  to  Liverpool, 
S6hhd.  ol'  tobacco,  each  weighing  450  pounds.  It  was  sold 
at  Liverpool  for  12^ri.  per  pound,  and  the  expenses  of  freight 
and  commissions  were  £92  I5.  8d.'  If  exchange  in  New 
York  is  at  a  premium  of  9^  per  cent,  what  should  the  ownei 
receive  for  the  bill  of  exchange,  in  United  States  Money  1 

276.  The  unit  or  base  of  the  French  Currency  is  the  French 
franc,  of  the  value  of  18  cents  6  mills.  The  franc  is  divided  into 
tenths,  called  decimes,  corresponding  to  our  dimes,  and  into 
centimes  corresponding  to  mills.  Thus,  5.12  is  read,  5  francs 
and  12  centimes. 

275.  What  is  a  foreign  bill  of  exchange  1  In  bills  on  England,  what 
is  the  unit  or  base  1  What  is  the  exchange  value  of  the  £  sterling '' 
How  much  is  the  true  value  above  the  commercial  value  of  the  £  ster- 
ling 1  How  do  you  find  the  value  of  a  bill  in  English  currency  in 
Uiiited  States  money  1 


26S  UUTfES. 

All  bills  of  exchange  on  France  are  drawn  in  francs.  Ex- 
change is  quoted  in  New  York  at  so  many  francs  and  centimes 
to  the  dollar. 

1.  What  will  be  the  value  of  a  bill  of  exchange  for  4536 
francs,  at  5.25  francs  to  the  dollar  ? 

Analysis.— Since  1  dollar  will  buy  operation 

5.25  francs,  the  bill  will  cost  as  many       ^  25)4536(^864  Arts. 
dollars  as  5.25  is  contained  times  in  the  ^  ^ 

amount  of  the  bill  :  hence, 

Divide  t}ce  amount  of  the  bill  by  the  value  of%l  i7t  francs  : 
the  quotient  is  the  amount  to  be  paid  in  dollars. 

2.  What  will  be  the  amount  to  be  paid,  United  States 
money,  for  a  bill  of  exchange  on  Paris,  of  6530  francs, — 
exchange  being  5.14  francs  per  dollar? 

3.  What  will  be  the  amount  to  be  paid  in  United  States 
money  for  a  bill  of  exchange  on  Paris  of  10262  francs,  ex- 
change being  5.09  francs  per  dollar  ? 

4.  Wliat  will  be  the  value  in  United  States  money  of  a 
bill  for  87595  francs,  at  5.16  francs  per  dollar  1 

DUTIES. 

277.  Persons  who  bring  goods  or  merchandise  into  the 
United  States,  from  foreign  countries,  are  required  to  land 
them  at  particular  places  or  Ports,  called  Ports  of  Entry,  and 
to  pay  a  certain  amount  on  their  value,  called  a  Duty.  This 
duty  is  imposed  by  the  General  Government,  and  must  be 
the  same  on  the  same  articles  of  merchandise,  in  every  part 
of  the  United  States. 

Besides  the  duties  on  merchandise,  vessels  employed  in 
commerce  are  required,  by  law,  to  pay  certain  sums  for  the 
privilege  of  entering  the  ports.  These  sums  are  large  or 
small,  in  proportion  to  the  size  or  tonnage  of  the  vessels. 
The  moneys  arising  from  duties  and  tonnage,  are  called 
revenues. 

276.  What  is  the  unit  or  base  of  the  French  currency  1  W'hat  is  its 
value  1  How  is  it  divided  1  In  what  currency  are  French  bills  of  ex- 
cliange  drawn  "? 

277.  What  is  a  port  entry  ''  W^hat  is  a  duty  1  By  whom  are  duties 
imposed  ^  What  charges  are  vessels  required  to  pay  1  What  are  the 
moneys  arising  from  duties  and  tonnage  called  ? 


DUTIKS.  269 

278.  Tlie  revenues  of  the  country  are  under  the  general 
direction  of  the  Secretary  of  the  Treasury,  and  to  secure  their 
faithful  collection,  the  government  has  appointed  various 
officers  at  each  port  of  entry  or  place  where  goods  may  be 
landed. 

279.  The  office  established  by  the  government  at  any  port 
of  entry  is  called  a  Custom  House,  and  the  officers  attached 
to  it  are  called  Custom  House  Officers. 

280.'  All  duties  levied  by  law^  on  goods  imported  into  the 
United  States,  are  collected  at  the  various  custom  houses,  and 
are  of  two  kinds,  Specific  and  Ad  valorem. 

A  specific  duty  is  a  certain  sum  on  a  particular  kind  of 
goods  named  ;  as  so  much  per  square  yard  on  cotton  or  wool- 
len cloths,  so  much  per  ton  weight  on  iron,  or  so  much  per 
gallon  on  molasses. 

An  ad  valorem  duty  is  such  a  per  cent  on  the  actual  cost 
of  the  goods  in  the  country  from  which  they  are  imported. 
Thus,  an  ad  valorem  duty  of  15  per  cent  on  English  cloths,  is 
a  duty  of  15  per  cent  on  the  cost  of  cloths  imported  from  Eng- 
and. 

281.  The  laws  of  Congress  provide,  that  the  cargoes  of  all 
vessels  freighted  with  foreign  goods  or  merchandise  shall  be 
weighed  or  gauged  by  the  custom  house  officers  at  the  port  to 
which  they  are  consigned.  As  duties  are  only  to  be  paid  on 
the  articles,  and  not  on  the  boxes,  casks  and  bags  which  con- 
tain them,  certain  deductions  are  made  from  the  weights  and 
measures,  called  Allowances. 

Gross  Weight  is  the  whole  weight  of  the  goods,  together 
with  that  of  the  hogshead,  barrel,  box,  bag,  &c.,  which  con- 
tains them. 

278.  Under  whose  direction  are  the  revenues  of  the  country  ? 

279.  What  is  a  custom  house  \  What  are  the  officers  attached  to  it 
called  1 

280.  Where  are  the  duties  collected  1  How  many  kinds  are  there, 
and  what  are  they  called  1  What  is  a  specific  duty  ^  An  ad  valorem 
duty  1 

281.  What  do  the  laws  of  Congress  direct  in  relation  to  foreign 
goods  I  \\'hy  are  deductions  made  from  their  weight  \  What  are 
these  deductions  called  !  What  is  gross  weight  \  W' hat  is  draft  ? 
What  is  the  greatest  draft  allowed  1  What  is  tare  !  What  are  the 
diflereiit  kinds  of  tare  1  -  What  allowances  are  made  on  liquors  1 


270  DUTIES. 

Draft  is  an  allowance  from  the  gross  weight  on  account  o! 
waste,  where  there  is  not  actual  tare. 

On     \\2lh,  it  is  Mh. 

From     112  to     224  "     2, 

224  to     336  "     3, 

336  to   1120  "     4, 

"      1120  to  2016  "     7, 

Above  2016  any  weight         "     9; 

fonseqiiently,  9Z6.  is  the  greatest  draft  allowed. 

Tare  is  an  allowance  made  for  the  weight  of  the  boxes, 
barrels,  or  bags  containing  the  commodity,  and  i&  of  three 
kinds  :  U/,  Legal  tare,  or  such  as  is  established  by  law  ;  2<:/, 
Customary  tare,  or  such  as  is  established  by  the  custom  among 
merchants  ;  and  3c/,  Actual  tare,  or  such  as  is  found  by  re- 
moving the  goods  and  actually  weighing  the  boxes  or  casks 
in  which  they  are  contained. 

On  liquors  in  casks,  customary  lure  is  sometimes  allowed 
on  the  supposition  that  the  cask  is  not  full,  or  what  is  called 
its  actual  wants ;  and  then  an  allowance  of  5  per  cent  for 
leakage. 

A  tare  of  10  per  cent  is  allowed  on  porter,  ale  and  beer,  in 
bottles,  on  account  of  breakage,  and  5  per  cent  on  all  other 
liquors  in  bottles.  At  the  custom  house,  bottles  of  the  com- 
mon size  are  estimated  to  contain  2|-  gallons  the  dozen. 

Note. — For  tables  of  Tare  and  Duty,  see  Ogden  on  the  Tariff 
ol"  1842. 

EXAMPLES. 

1.  What  will  be  the  duty  on  125  cartons  of  ribbons,  each 
containing  48  pieces,  and  each  piece  w^eighing  Zoz.  net,  and 
paying  a  duty  of  |2,50  per  pound  ? 

2.  W%at  will  be  the  duty  on  225  bags  of  coffee,  each  weigh- 
ing gross  160/6.,  invoiced  at  6  cents  per  pound  ;  2  per  cent 
being  the  legal  rate  of  tare,  and  20  per  cent  the  duty  ? 

3.  What  duty  must  be  paid  on  275  dozen  bottles  of  claret, 
estimated  to  contain  2|  gallons  per  dozen,  5  per  cent  being 
allowed  for  breakage,  and  the  duty  being  35  cents  per  gallon  '] 

4.  A  merchant  imports  175  cases  of  indigo,  each  case 
weighing  196/6.S'.  gross  ;  15  per  cent  is  the  customary  rate  of 
tare,  and  the  duty  5  cents  per  pound  :  what  duty  must  he 
pay  on  the  whole  ? 


ALLIGATION   MEDIAL.  271 


ALLIGATION   MEDIAL. 

28.2.  Alligation  Medial  is  the  process  of  finding-  the 
price  of  a  mixture  when  the  quantity  of  each  simple  and  it? 
price  are  known. 

1.  A  merchant  mixes  Sib.  of  tea,  worth  75  cents  a  pound 
with  161b.  worth  ^1,02  a  pound  :  what  is  the  price  of  the 
mixture  per  pound  '^ 

Analysis. — The  quantity,  8/6.  of  operation. 

tea,  at  75  cents  a  pound,  costs  $6;  6Ib.  at  '75cts.z=$  6,00 
and  16/6.  at  Sl.02  costs  $16,32:  16/^.  at  $1,02  =  $16,32 
hence,    the    mixture.  =  24/6.,    costs      ^r —  oas^o  ^o 

$32,32  ;  and  the  price  of  1/6.  of  the       ^*  "^^^'^^ 

mixture  is   found    by  dividing  this  $0,93 

cost  by  24 :  hence,  to  find  the  price  of  the  mixture, 

I.  Find  the  cost  of  the  entire  mixture  : 

II.  Divide  the  entire  cost  of  the  mixture  by  the  sum  of 
the  simples,  and  the  quotient  toill  be  the  jynce  of  the  mixture. 

examples. 

1.  A  farmer  mixes  30  bushels  of  wheat  worth  55.  per 
bushel,  with  72  bushels  of  rye  at  35.  per  bushel,  and  with 
60  bushels  of  barley  worth  2s.  per  bushel  :  what  should  be 
the  price  of  a  bushel  of  the  mixture  ? 

2.  A  wine  merchant  mixes  15  gallons  of  wine  at  $1  per 
gallon  with  25  gallons  of  brandy  worth  75  cents  per  gallon  : 
what  should  be  the  price  of  a  gallon  of  the  compound  ? 

3.  A  grocer  mixes  40  gallons  of  whisky  worth  31  cents 
per  gallon  with  3  gallons  of  M'ater  which  costs  nothing  :  what 
should  be  the  price  of  a  gallon  of  the  mixture? 

4.  A  goldsmith  melts  together  2/6.  of  gold  of  22  carats 
fine,  602.  of  20  carats  fine,  and  602.  of  16  carats  fine  :  what 
is  the  fineness  of  the  mixture  ? 

5.  On  a  certain  day  the  mercury  in  the  thermometer  was 
observed  to  average  the  following  heights :  from  6  in  the 
morning  to  9,  64°  ;  from  9  to  12,  74°  ;  from  12  to  3,  81°  ; 
and  from  3  to  6,  70°  :  what  was  the  mean  temperature  of 
the  day  ? 

'^82.  What  is  Alligation  Medial  1  What  is  the  rule  for  determhiing 
the  price  of  the  mixture  1 


272 


ALIJGATION    ALTICRNATE. 


A. 

B. 

C. 

D. 

E. 

i 

^ 

2 

1 

3 

1 

2 

2 

J. 
4 

1 

1 

ALLIGATION    ALTERNATE. 

283.  Alligation  Alternate  is  the  process  of  firuliiio-  wha; 
proportions  must  be  taken  of  each  of  several  simples,  whos€ 
prices  are  known,  to  form  a  compound  of  a  ^riven  price.  Ti 
is  the  opposite  of  Alligation  Medial,  and  may  be  proved  by  it. 

284.  To  find  the  'proportional  parts  : 

1.  A  farmer  would  mix  oats  at  35.  a  bushel,  rye  at  6v.,  and 
wheat  at  9.s'.  a  bushel,  so  that  the  mixture  shall  be  worth  5 
shillings  a  bushel  :  what  proportion  must  be  taken  of  each 
Bort? 

OPERATION 


oats,  3-1 
rye,  GJ 
wheat,  9 

Analysis. — On  e/ery  bushel  put  into  the  mixture,  whose  price 
is  less  than  the  mean  price,  there  will  be  a  gain;  on  every  bushel 
whose  price  is  greater  than  the  mean  price,  there  will  be  a  /o.v.s  ; 
and  since  there  is  to  be  neither  gain  nor  loss  by  the  mixture,  the 
gains  and  losses  must  balance  each  other. 

A  bushel  of  oats,  when  put  into  the  mixture,  will  bring  5  shil- 
lings, giving  a  gain  of  2  shillings  ;  and  to  gain  1  shilling,  we  must 
take  half  as  much,  or  -^  a  bushel,  which  we  write  in  column  A. 

On  1  bushel  of  wheat  there  will  be  a  loss  of  4  shillings ;  and 
to  make  a  loss  of  1  shilling,  we  must  take  |-  of  a  bushel,  which 
we  also  write  in  column  A  :  -J  and  ^  are  called  proportional 
numbers. 

Again  :  comparing  the  oats  and  rye.  there  is  a  gain  of  2  shil- 
lings on  every  bushel  of  oats,  and  a  loss  of  1  shilling  on  every 
bushel  of  rye:  to  gain  1  shilling  on  the  oats,  w^e  take  ^  a  bushel, 
and  to  lose  1  shilling  on  the  rye,  we  take  1  bushel  :  these  num- 
bers are  written  in  column  B.  Two  simples,  thus  compared,  are 
called  a  couplet:  in  one.  the  price  of  unity  is  less  than  the  mean 
piice,  and  in  the  other  it  is  greater. 

If,  every  time  we  take  ^  a  bushel  of  oats  we  take  ^  of  a  bushel 
of  wheat,  the  gain  and  loss  will  balance;  and  if  every  time  we 
take  -^  a  bushel  of  oats  we  take  1  bushel  of  rye.  the  gain  and  loss 


283.  What  is  Alligation  Alternate  ? 

284.  How  do  you  find  the  proportional  numbers  1 


ALLIGATION    ALTERNATE. 


273 


Will  balance  :  hence,  if  the  proportional  numbers  of  a  couplet  be 
multiplied  by  any  number^  the  gain  and  loss  denoted  by  the  products, 
will  balance. 

When  the  proportional  numbers,  in  any  column,  are  fractional 
(as  in  columns  A  and  B),  multiply  them  by  the  least  common 
divi.sor  of  their  denominators,  and  write  the  products  in  new- 
columns  C  and  D.  Then,  add  the  numbers  in  columns  C  and  D, 
standing  opposite  each  simple,  and  if  their  sums  have  a  common 
factor,  reject  it :  the  last  result  will  be  the  proportional  numbers, 

Rule. — I.  Write  the  prices  or  qualities  of  the  simples  in  a 
column,  beghming  with  the  lowest,  and  the  mean  price  or 
quality  at  the  left, 

II.  Opposite  the  first  simple  write  the  part  which  must  he 
taken  to  gain  1  of  the  mean  price,  and  opposite  the  other  simple 
of  the  couplet,  write  the  part  which  juust  he  taken  to  lose  1  of 
the  mea.7i  price,  and  do  the  same  for  each  simple. 

III.  When  the  proportional  numbers  are  fractional,  reduce 
them  to  integral  numbers,  and  then  add  those  which  stand  oppo- 
site the  same  sifnple  :  if  the  sums  have  a  common  factor,  reject 
it:  the  result  will  denote  the  proportional  parts. 

2.  A  merchant  would  mix  wines  worth  IGs.,  ISs.,  and  22s. 
per  gallon,  in  such  a  way,  that  the  mixture  may  be  w^rth 
20s.  per  gallon  :  what  are  the  proportional  parts  1 

OPERATION. 

A. 

20  J  18-, 
(22J 

PROOF. 

1  gallon,  at  16  shillings,  =  16s. 

1  gallon,  at  18  shillings,  =  18s. 

3  gallon,  at  22  shillings,  =  66s. 

5)100(20s.,  mean  price. 

Note. — The  answers  to  the  last,  and  to  "all  similar  questions, 
will  be  infinite  in  number,  for  two  reasons  : 

Ist.  If  the  proportional  numbers  in  column  E  be  multiplied  by 
any  number,  integral  or  fractional,  the  products  will  denote  pro- 
tiortional  parts  of  the  simples. 

2d.  If  the  proportional  numbers  of  ayiy  touyUt  be  muUiplied  by 


A. 

B. 

C. 

D. 

E. 

i 

1 

1 

i 

1 

1 

i 

i 

2 

1 

3 

274:  ALLIGATION    ALTERNATE. 

any, number,  the  gain  and  loss  in  that  couplet  will  still  balancCj 
arid  ,the  proportional  numbers  in  the  final  result  will  be  changed. 

3.  What  proportions  of  tea,  at  24  cents,  30  cents,  33  cents 
and  36  cents  a  pound,  must  be  mixed  together  so  that  the 
mixture  shall  be  worth  32  cents  a  pound  ? 

4.  What  proportions  of  coffee  at  IQcts.,  20cts.  and  28cf,s. 
per  pound,  must  be  mixed  together  so  that  the  compound 
shall  be  worth  24:Ct.s.  per  pound  ? 

5.  A  goldsmith  has  gold  of  16,  of  18,  of  23,  and  of  24  carats 
fine  .  what  part  must  be  taken  of  each  so  that  the  mixture 
shall  be  21  carats  fine  ? 

6.  What  poition  of  brandy,  at  14s.  per  gallon,  of  old  Ma 
deira,  at  24s.  per  gallon,  of  new  Madeira,  at  21s.  per  gallon, 
and  of  brandy,  at  10s.  per  gallon,  must  be  mixed  together  so 
that  the  mixture  shall  be  worth  1 8s.  per  gallon  1 

285.    When  the  quantity  of  one  sitnple  is  given  : 

1.  How  much  wheat,  at  9s.  a  bushel,  must  be  mixed  with 
20  bushels  of  oats  w^orth  3  shillings  a  bushel,  that  the  mix- 
ture may  be  worth  5  shillings  a  bushel  ? 

Analysis. — Find  the  proportional  numbers  :  they  are  2  and  1 ; 
hence,  Ihe  ratio  of  the  oats  to  the  wheat  is  ^ :  therefore,  there 
must  be  10  bushels  of  wheat. 

Rule. — I.  Find  the  2^foportional  numbers^  and  write  the 
given  simple  opposite  its  proportional  number. 

II.  Muliiply  the  given  simiole  by  the  ratio  which  its  proper' 
tional  number  bears  to  each  of  the  others,  and  the  products 
will  denote  the  quantities  to  be  taken  of  each, 

EXAMPLES. 

1.  How  much  wine,  at  5s.,  at  5s.  6d.,  and  6s.  per  gallon 
must  be  mixed  with  4  gallons,  at  4s.  per  gallon,  so  that  the 
mixture  shall  be  worth  5s.  4d.  per  gallon  ? 

2.  A  farmer  would  mix  14  bushels  of  wheat,  at  $1,20  per 
bushel,  with  rye  at  12cts.,  barley  at  486'^s.,  and  oats  at  36rfs.  : 
how  much  must  be  taken  of  each  sort  to  make  the  mixture 
worth  64  cents  per  bushel  1 

3.  There  is  a  mixture  made  of  wheat  at  4s.  per  bushel, 
rye  at  3s.,  barley  at  2s.,  \Vith  12  bushels  of  oats  at  \Bd.  pei 
bushel  :  how  much  is  taken  of  each  sort  when  the  mixture  is 
worth  3 5.  6^/.  '\ 


ALLIGATION   ALTER 

4.   A  distiller  would  mix  iOgal.  of 
per  gallon,  \vith  English  at   7^.  and  spiril 
what  quantity  must  be  taken  of  each  sort 
may  be  afforded  at  85.  per  gallon? 

286.    When  the  quantity  of  the  mixture  is  given. 

I.  A  merchant  would  make  up  a  cask  of  wine  containing 
60  gallons,  with  wine  worth  16^.,  I8s.  and  226'.  a  gallon,  in 
such  a  way  that  the  mixture  may  be  worth  20s.  a  gallon  • 
how  much  must  he  take  of  each  sort  1 

Analysis. — This  is  the  same  as  example  2,  except  that  the 
quantity  of  the  mixture  is  given.  If  the  quantity  of  the  mixture 
be  divided  by  5,  the  sum  of  the  proportional  parts,  the  quotient 
10  vsi  11  show  how  many  times  each  proportional  part  must  be  taken 
to  make  up  50  gallons:  hence,  there  are  10  gallons  of  the  first, 
10  of  the  second,  and  30  of  the  third  :  hence. 

Rule. — I.   Find  the  jM-oporttonal  parts. 

II.  Divide  the  quantity  of  the  mixture  by  the  sum  of  the 
proportional  parts,  and  the  quotient  vnll  denote  how  many 
titnes  each  part  is  to  he  taken.  Multiply  this  quotient  by 
the  parts  separately,  and  each  product  vnll  denote  the  quan- 
tity of  the  corresponding  simple. 

EXAMPLES. 

1.  A  grocer  has  four  sorts  of  sugar,  worth  12cZ.,  lOo?.,  6c?. 
and  4c?.  per  pound  ;  he  would  make  a  mixture  of  144  pounds 
worth  8c/.  per  pound  :  what  quantity  must  be  taken  of  each 
sort '? 

2.  A  grocer  having  four  sorts  of  tea,  worth  5s.,  6s.,  8s  and 
96-.  per  pound,  wishes  a  mixture  of  87  pounds  worth  7s.  per 
pound  :  how  much  must  he  take  of  each  sort  ? 

3.  A'  silversmith  has  four  sorts  of  gold,  viz.,  of  24  carats 
fine,  of  22  carats  fine,  of  20  carats  fine,  and  of  15  carats  fine ; 
he  would  make  a  mixture  of  A2oz.  of  17  carats  fine;  hov.' 
much  must  be  taken  of  each  sort  % 

Proof. — All  the  examples  of  AUigation  Medial  may  be 
found  by  Alligation  Alternate. 

285.  How  do  you  find  the  quantity  of  each  simple  when  the  quantity 
of  one  simple  is  known  1 

286.  How  do  you  find  the  quantity  of  eac-li  simple  when  the  quaniitj 
tif  each  mixture  i?  known  ' 


276  INVOLUTION. 

INVOLUTION. 

287.  A  POWER  is  the  product  of  equal  factors.  The  equal 
factor  is  called  the  root  of  the  power. 

The  first  power  is  the  equal  factor  itself,  or  the  root : 
The  second  power  is  the  product  of  the  root  by  itself  : 
The  third  power  is  the  product  when  the  root  is  taken  3 
times  as  a  factor  : 

The  fourth  power,  when  it  is  taken  4  times  : 
The  fifth  power,  when  it  is  taken  5  times,  &c. 

288.  The  number  denoting  how  many  times  the  root  is 
taken  as  a  factor,  is  called  the  exponent  of  the  power.  It  is 
written  a  little  at  the  right  and  over  the  root  :  thus,  if  the 
equal  factor  or  root  is  4. 

4=       4  the  1st  power  of  4. 

42  =  4x4=      16  the  2d  power  of  4. 

4^  =  4x4x4=      64  the  3d  power  of  4. 

4*  ==4x4x4x4=   256  the  4th  power  of  4. 

4^  =  4x4x4x4x4=1024  the  othpowerof4. 

Involution  is  the  process  of  finding  the  powers  of , lumber 8. 

Notes. — 1.  There  are  three  things  connected  with  every  power  • 
1st,  The  root ;  2d3  The  exponent  ]  and  Sd,  The  power  or  result  of 
the  multiplication. 

2.  In  finding  a  power,  the  root  is  always  the  1st  power  :  hence, 
the  number  of  multiplications  is  1  less  than  the  exponent : 

Rule. — Multiply  the  number  by  itself  as  many  times  less 
1  as  there  are  units  in  the  exponent^  and  the  last  jn'oduct 
will  be  the  power. 

EXAMPLES. 


Find  the  powers  of  the  following  numbers  : 

10.  5th  power  of  16. 

11.  6th  power  of  20. 

12.  2d    power  of  225. 


1.  Square  of  1. 

2.  Square  of  ^. 

3.  Cube  of  f 

4.  Square  ol"  j. 

5.  Square  of  9. 

6.  Cube  of  12. 

7.  3d    power  ol  125, 

8.  3d    power  of  16. 
y.  4  th  power  of  9. 


13.  Square  of  2167. 

14.  Cube  of  321. 

15.  4th  power  of  215. 

16.  5th  power  of  906. 

17.  6th  power  of  9. 
IS.   ;^<iuare  of  3GU'i9. 


EVOLUTION.  27 1 

EVOLUTION. 

289.  Evolution  is  the  process  of  finding  the  factor  when 
we  know  the  power. 

The  square  root  of  a  number  is  the  factor  which  niuhiphed 
by  itself  once  will  produce  the  number. 

The  cube  root  of  a  number  is  the  factor  which  multiplied 
y  itself  twice  will  produce  the  number. 

Thus,  6  is  the  square  root  of  36,  because  6  X  6  =  36  ;  and 
3  is  the  cube  root  of  27,  because  3  X  3  X  3  =  27. 

The  sign  -y/  is  called  the  radical  sign.  "When  placed  be- 
fore a  number  it  denotes  that  its  square  root  is  to  be  ex- 
tracted.    Thus,  y"36  =  6. 

We  denote  the  cube  root  by  the  same  sign  by  writing  3 
over  it  :  thus,  \/27  denotes  the  cube  root  of  27,  which  is 
eqiial  to  3.  The  small  figure  3,  placed  over  the  radical,  is 
called  the  index  of  the  root. 

EXTRACTION  OF  THE  SQUARE  ROOT. 

290.  The  square  root  ol  a  number  is  a  factor  which  mul- 
tiplied by  itself  once  will  produce  the  number.  To  extract 
the  square  root  is  to  find  this  factor.  The  first  ten  numbers 
and  their  squares  are 

1,      2,      3,       4,        5,        6,        7,        8,         9,         10. 
1,       4,       9,       16,       25,       36,      49,       64,       81,       100. 

The  numbers  in  the  first  line  are  the  square  roots  of  those 
in  the  second.  The  numbers  1,  4,  9,  16,  25,  36,  &c. 
having  exact  factors,  are  called  perfect  squares. 

A  perfect  square  is  a  number  which  has  two  exact  factors. 

Note. — The  square  root  of  a  number  less  than  100  will  be  less 
tha.^  10,  while  the  square  root  of  a  number  greater  than  100  will 
be  "i^ater  than  10. 


287.  "'Vhat  is  a  power  1  What  is  the  root  of  a  power  1  What  is  the 
friSt  powfc-  I     What  is  the  second  power  !     The  third  power  ] 

i.S8.  Wl.Ht  is  the  exponent  of  the  power  1  How  is  it  written  ]  ^^  hat 
is  In  fi)lution  1  How  many  things  are  connected  with  every  power'' 
How  v'o  you  find  the  power  of  a  number? 

289.  What  is  Evolution  1  Vl'hat  is  the  square  root  of  a  number  ' 
What  is  the  cube  root  of  a  number  1  How  do  you  denote  the  f-»jtiarc 
loot  of  a  MUJiibor'?     How  the  cube  loot  ? 


278 


EXTRACTION   OF   THE   SQUAKE   EOOT. 


291.  What  is  the  square  of  3 6  ==3  tens +6  units? 


Analysis. — 36=3  tens +6  units,  is  first 


3+6 

3  +  f> 


3x6- 
32  +  3x6 


30 


to  be  taken  6  units'  times,  giving  6^  +  3X6: 
then  taking  it  3  tens'  times,  we  have 
3 X6-t-32,  and  the  sum  is  32+2(3x6) +  62: 

^^'^^'"'  32  +  2(3x6)  +  6'-^ 

The  square  of  a  number  is  equal  to  the  square  of  the  tens, 

plus  twice  the  product  of  the  tens  by  the  units,  plus  the  square 

of  the  units. 

The  same  may  be  shown  by  the  figure : 

Let  the  hne  AB  re- 
present the  3  tens  or  30, 
and  BC  the  six  units. 

Let  AD  be  a  square 
on  AC,  and  AE  a  square 
on  the  ten's  line  AB. 

Then  ED  will  be  a 
square  on  the  unit  line 
6,  and  the  rectangle  EF 
will  be  the  product  of 
HE,  which  is  equal  to 
the  ten's  line,  by  IE, 
which  is  equal  to  the 
unit  line.  Also,  the 
rectangle  BK  will  be  the 

product  of  EB,  which  is     a  30  B  t; 

equal  to  the  ten's  line,  by 

the  unit  line  B  C.  But  the  whole  square  on  AC  is  made  up  of 
the  square  AE,  the  two  rectangles  FE  and  EC,  and  the  square 
ED. 

1.  Let  it  now  be  required  to  extract  the  square  root  of 
1296. 

Analysis. — Since  the  number  contains  more  than  two  places  ot 
figures,  its  root  will  contain  tens  and  units.  But  as  the  square  of 
one  ten  is  one  hundred,  it  follows  that  the  square  of  the  tens  of 
the  required  root  must  be  found  in  the  two  figures  on  the  left  ol 
96.  Hence,  we  point  off  the  number  into  periods  of  two  figures 
each. 

290.  "What  is  the  square  root  of  a  number?  What  are  perfect 
squares  \     How  many  are  there  between  1  and  100  1 

291.  Into  what  parts  may  a  number  be  dcnonijmiied  1  Whou  so  dc- 
Lomposod.  wliut  is  its  S(|u;ue  rqiiiil  to  ! 


30 

6 

6 

6 

180 

36 

K 

30       E 

900  +  180  +  180+36  =  1296. 

30 

30 

S 

30 

6 

900 

180 

EXTUACTION    OF   THE    SQUARE   liOOT.  279 

We  next  find  the  greatest  square  contained  in  operation. 

12,  which  is   3  tens  or   30.     We  then  square   3  |2  96(36 

tens  which  gives  9  hundred,  and  then  place  9  un-  g 

der  the  hundreds'  place,  and  subtract  ;  this  takes 

away  the   square   of    the   tens,  and    leaves   396,  66)396 

which   is  twice  the  product  of  the  tens  by  the  units  396 
•plus  the  square  of  the  units. 

If  now,  we  double  the  divisor  and  then  divide  this  remainder, 
exclusi^De  of  the  right  hand  figure,  (since  that  figure  cannot  enter 
into  the  product  of  the  tens  by  the  units)  by  it,  the  quotient  will 
be  the  units  figure  of  the  root.  If  we  annex  this  figure  to  the 
augmented  divisor,  and  then  multiply  the  whole  divisor  thus  in- 
creased by  it,  the  product  will  be  twice  the  tens  by  the  units  plus 
the  square  of  the  units  ;  and  hence,  we  have  found  both  figures  of 
ihe  root. 

This  process  may  also  be  illustrated  by  the  figure. 

Subtracting  the  square  of  the  tens  is  taking  away  the  square 
AE  and  leaves  the  two  rectangles  FE  and  BK,  together  with  the 
square  ED  on  the  unit  line. 

The  two  rectangles  FE  and  BK  representing  the  product  of  units 
by  tens,  can  be  expressed  by  no  figures  less  than  tens. 

If,  then,  we  divide  the  figures  39,  at  the  left  of  6,  by  twice  the 
tens,  that  is,  by  twice  AB  or  BE,  the  quotient  will  be  BC  or  EK, 
the  unit  of  the  root 

Then,  placing  BC  or  6,  in  the  root,  and  also  annexing  it  to  the 
divisor  doubled,  and  then  multiplying  the  whole  divisor  66  by  6, 
we  obtain  the  two  rectangles  FE  and  CE.  together  with  the 
square  ED. 

292.  Hence,  for  the  extraction  of  the  square  root,  we  have 
the  following 

E.ULE. — I.  Separate  the  given  number  into  periods  of  two 
figures  each,  by  setting  a  dot  over  the  place  of  units,  a  se- 
cond over  the  place  of  hundreds,  and  so  on  for  each  alternate 
figure  at  the  left. 

II.  Note  the  greatest  square  contained  in  the  period  on 
ihe  left,  and  place  its  root  on  the  right  after  ihe  manner  of 
a  quotient  tn  division.  Subtract  the  square  of  this  root 
from  the  first  period,  md  to  the  remainder  bring  down  the 
secmid  period  for  a  dividend. 

292.  What  is  the  first  step  in  extracting  the  square  root  of  numbers  1 
What  is  the  secoivJ  (  What  is  the  third  I  What  the  fourth  ?  What 
Ihe  fifth  1     (Jive  the  entire  rule 


280  EXTiiAcnoN  OF  Till;  sc^uark  root. 

III.  Double  the  root  thus  found  for  a  trial  divisor  and 
•place  it  on  the  left  of  the  dividend.  Find  how  many 
times  the  trial  divisor  is  contained  in  the  dividend,  exclu- 
sive of  the  right-hand  figure,  and  place  the  quotient  in  tit^ 
root  and  also  a?inex  it  to  the  divisor. 

IV.  Midtij)ly  the  divisor  thus  increased,  by  the  last  figure 
of  the  root ;  subtract  the  product  from,  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  divi^ 
dend. 

V.  Double  the  whole  root  thus  found,  for  a  new  trial  di- 
visor, and  continue  the  operation  as  before,  until  all  the 
periods  are  brought  down. 

EXAMPLES. 

1.  What  is  the  square  root  of  263169  % 
Analysis^. — We  first,  place  a  dot  over  the  operation. 

9,  making  the  riglit-hand  period  69.     We  26   Si    69(513 

then  put  a  dot  over  the  1  and  also  over  the  «< 

6,  making  three  periods. 


The  greatest  periect  square  in  26  is  25,       101)1^V 
the  root  of  whicli  is  5.     Placing  5  in  the  lOl 

root,  sublracting  its  square  from    26,  and       1023  ")30(')I)  ' 
bringing  down  the  next  period  31,  we  have  SO  (9 

131    for   a  dividend,  and    by  doubling  the 

root  v^-e  have  10  for  a  trial  divisor.  Now.  10  is  contained  in  13, 
1  time.  Place  1  both  in  the  root  and  in  the  divisor  :  ii.cn  multi- 
ply 101  by  1  ;  subtract  the  product  and  bring  down  the  lext  period. 

We  must  now  double  the  whole  root  51  for  a  new  trial  divisor  ; 
or  we  may  take  the  first  divisor  after  havfng  doubled  the  last 
figure  1  ;  then  dividing,  we  obtain  3,  the  third  figure  of  the  root. 

Notes. — 1.  The  left-hand  period  may  contain  but  one  figure  ; 
each  of  the  others  will  contain  two. 

2.  If  any  trial  divisor  is  greater  than  its  dividend,  the  corres- 
ponding quotient  figure  will  be  a  cipher. 

3.  If  the  product  of  the  divisor  by  any  figure  of  the  root  exceeds 
the  corresponding  dividend,  the  quotient  figure  is  too  large  and 
muft  be  diminished.    . 

4.  There  will  be  as  many  figures  in  the  root  as  there  are  periods 
in  the  given  number. 

5.  If  the  given  number  is  not  a  perfect  square  there  will  be  a 
remainder  after  all  the  periods  are  brought  down.  In  this  case, 
periods  of  ciphers  may  be  annexed,  forming  new  periods,  each  of 
V'liich  will  give  one  decimal  place  iuthe  root. 


EXl'BAOTION    OF    TLHO    SQTTAKK    liOOT. 


281 


2    Wliat  ig  the  square  root  of  36729?       operation. 


Ill  this  example  there  are  two 
periods  of  decimals,  which  give  two 
places  ef  decimals  in  the  root. 


3  67  29(191  64-f-, 

1 


29)267 


261 


381)629 
381 
3826)24800 
22956 


38324)184400 
153296 


31104  Rem 


293.    To  extract  the  square  root  of  a  fractiim 


1.  What  is  the  square  root  of  .5  1 


Note. — We  first  annex  one  cipher  to 
make  even  decimal  places.  We  then  ex- 
tract the  root  of  the  first  period  :  to  the 
remainder  we  annex  two  ciphers,  forming 
a  new  period,  and  so  on. 


OPERATION. 

.60(.707-|- 
49 


140)100 
000 


1407)10000 
9849 


151   Renu 


2.  What  is  the  square  root  of  J  ? 

Note. — The  square  root  of  a  fraction 
is  equal  to  the  square  root  of  the  numerator 
divided  by  the  square  root  of  the  denomi- 
nator. 

3.  What  is  the  square  root  of  J  ? 

Note. — When  the  terms  are  not  per- 
fect squares,  reduce  the  common  fraction 
to  a  decimal  fraction,  and  then  extract 
the  square  root  of  the  decimal. 


OPERATION 

'4 


v/: 


v/4_2 


OPERATION. 

:.75; 


v/|=V?75=..854d-|- 


29'J.  How  do  you  extract  the  w^uare  root  of  a  deciuial  fraction  1 
uf ;.  cumumui  fraction  ! 

10 


IJow 


282 


SQUAKE   KOOT. 


Rule. — 1.  If  the  fraction  is  a  decimal,  j^oint  off  tft^ 
periods  from  the  decimal  point  to  the  right,  annexing  ci- 
phers if  necessary,  so  that  each  period  shall  co7itain  two 
places,  and  then  extract  the  root  as  in  integral  numbers. 

II.  If  the  fraction  is  a  common  fraction,  and  its  terms 
perfect  squares,  extract  the  square  root  of  the  numerator  and 
denominator  separately  ;  if  they  are  not  perfect  squares,  re- 
duce the  fraction  to  a  decimal,  and  then  extract  the  square 
root  of  the  result, 

EXAMPLES. 

What  are  the  square  roots  of  the  following  numbers? 


1. 

of  3"? 

2. 

of  lU 

3. 

of  10691 

4. 

of  2268741 ? 

5. 

of  7596796  1 

6. 

of  36372961  ? 

7. 

of  22071204? 

8. 

of  3271.4207  ? 

9. 

of  4795.25731  ? 

10. 

of  4.372594  ? 

11. 

of  .0025? 

12. 

of  .00032754? 

13. 

of  .00103041 ? 

14. 

of  4.426816? 

15. 

of  8|? 

16. 

of  9  J  ? 

17. 

18. 
19. 
20. 

of /A? 
ofi|f1 

APPLICATIONS    IN    SQUARE    ROOT. 

294.   A  triangle  is  a  plain  figure  which  has  three  sides  and 
three  angles. 


If  a  straight  line  meets  another  straight  line, 
making  the  adjacent  angles  equal,  each  is 
called  a  right  angle  ;  and  the  lines  are  said 
to  be  perpendicular  to  each  other. 


295.  A  right  angled  triangle  is  one 
which  has  one  right  angle.  In  the  right 
angled  triangle  ABC,  the  side  AC  opposite 
the  right  angle  B  is  called  the  hypothenuse  ; 
the  side  AB  the  base;  and  the  side  BC 
tlie  perpendicular 


AI'FLICATIONS. 


2S? 


296.  In  a  right  angled  triangle  the  square  described  in  the 
hypothenuse  is  equal  to  the  sum  of  the  squares  described  in 
the  other  two  sides. 

Thus,  if  ACB  be  aright 
angled  triangle,  right  an- 
gled at  C,  then  will  the 
large  square,  D,  described 
in  the  hypothenuse  AB,  be 
equal  to  the  sum  of  the 
squares  F  and  E  described 
on  the  sides  AC  and  CB. 
This  is  called  the  carpen- 
ter's theorem.  By  count- 
ing the  small  squares  in  the 
large  square  D,  you  will 
find  their  number  equal 
to  that    contained    in    the 

small  squares  F  and  E.  In  this  triangle  the  hypothenuse 
AB:^5,  AC =4,  and  CB  =  3.  Any  numbers  having  the 
same  ratio,  as  5,  4  and  3,  such  as  10,  8  and  6  ;  20,  16  and 
12,  &c.,  will  represent  the  sides  of  a  right  angled  triangle. 

I.  Wishing  to  know  the  distance  from  A 
to  the  top  of  a  tower,  I  measured  the  height 
of  the  tower  and  found  it  to  be  40  feet  ;  also 
the  distance  from  A  to  B  and  found  it  30  feet; 
what  was  the  distance  from  A  to  C  ^ 

0  =  30  ;  AB2  =  302=:   900 

BC=40  ;  BC2=402  =  1600 

AC^^AB^+BC^zzz       2500 

AC=:V2500  =  50  feet. 


iB 

D 

297.  Hence,  when  the  base  and  perpendicular  are  known 
•lid  the  hypothenuse  is  required. 


294 

295 


Wliat  is  a  triangle  1     What  is  a  right  angle  1 

What  is  a  right  angled  triangle  1     Which  side  is  the  hypotlie- 


296.  In  a  right  aiigled   triangle  wliat   i?  the  square  on   the   hypothe- 
aU€e  equul  tui 


284  t;QUARE    ROOT. 

Square   the  base  and  square  the  perpendicular,  add  the  t- 
suits  and  then  extract  the  .square  root  of  their  sum. 

2.  What  is  the  length  of  a  rafter  that  will  reach  from  th 
eaves  to  the  ridge  pole  of  a  house,  when  the  height  of  the 
roof  is  15  feet  and  the  width  of  the  building  40  feet  ? 

298.  i'o  find  one  side  when  we  know  the  hypothenuse  and 
the  other  side. 

3.  The  length  of  a  ladder  which  will  reach  from  the  mid- 
dle of  a  street  80  feet  wide  to  the  eaves  of  a  house,  is  50  feet : 
what  is  the  height  of  the  house  ? 

Analysis. — Since  the  square  of  the  length  of  the  ladder  is  equal 
to  the  sum  of  the  squares  of  half  the  street  and  the  height  of  the 
house,  the  square  of  the  length  of  the  ladder  diminished  by  the 
square  of  half  the  street  will  be  equal  to  the  square  of  the  height 
of  the  house  :  hence, 

Square  the  hypothenuse  and  the  known  side,  and  take  the 
difference  ;  the  squure  root  of  the  difference  will  be  the  other 
side, 

EXAMPLES. 

1.  If  an  acre  of  land  be  laid  out  in  a  square  form,  what 
will  be  the  length  of  each  side  in  rods  ? 

2.  What  will  be  the  length  of  the  side  of  a  square,  in  rods, 
that  shall  contain  100  acres? 

3.  A  general  has  an  army  of  7225  men  :  how  many  must 
be  put  in  each  line  in  order  to  place  them  in  a  square  form  1 

4.  Two  persons  start  from  the  same  point ;  one  travel? 
due  east  50  miles,  the  other  due  south  84  miles  :  how  far  are 
they  apart  ? 

5.  What  is  the  length,  in  rods,  of  one  side  of  a  square  that 
shall  contain  12  acres  ? 

6.  A  company  of  speculators  bought  a  tract  of  land  foi 
^6724,  each  agreeing  to  pay  as  many  dollars  as  there  were 
partners  :  how  many  partners  were  there  ? 


297.  How  do  you  find  the  hypothenuse  when  you  know  the  base 
and  perpendicular  ! 

298.  If  you  know  the  hyputh^iuise  and  cue  side,  how  il'i  vou  finiJ  '.c 
othr  r  sic]«>  ■ 


CVBK    ROOT.  285 

7.  A  farmer  wishes  to  set  out  an  orchard  of  3844  trees,  so 
that  the  number  of  rows  shall  be  equal  to  the  number  of 
trees  in  each  row  :  what  will  be  the  number  of  trees  ? 

8.  How  many  rods  of  fence  will  enclose  a  square  field  of 
10  acres  ? 

9.  If  a  line  150  feet  long  will  reach  from  the  top  of  a 
steeple  120  feet  high,  to  the  opposite  side  of  the  street,  what 
is  the  width  of  the  street  ? 

1^0.  What  is  the  length  of  a  brace  whose  ends  are  each  3^ 
feet  from  the  angle  made  by  the  post  and  beam  ? 

CUBE    ROOT. 

299.  The  Cube  Root  of  a  number  is  one  of  three  equal 
factors  of  the  number. 

To  extract  the  cube  root  of  a  number  is  to  find  a  factor 
which  multiphed  into  itself  twice,  will  produce  the  given 
number. 

Thus,  2  is  the  cube  root  of  8  ;  for,  2x2x2  =  8:  and  3  is 
the  cube  root  of  27  ;  for  3  X  3  X  3  =  27. 

1,      2,       3,         4,         5,  6,  7,  8,  9. 

I   8   27   64   125   216   343   512   729 

The  numbers  in  the  first  line  are  the  cube  roots  of  the 
corresponding  numbers  of  the  second.  The  numbers  of  the 
second  line  are  called  perfect  cubes.  By  examining  the  num- 
bers of  the  two  lines  we  see, 

1st.  That  the  cube  of  units  cannot  give  a  higher  order  than 
hundreds. 

2d.  That  since  the  cube  of  one  ten  (10)  is  1000  and  the 
cube  of  9  tens  (90),  81000,  the  cube  of  tens  will  not  give  a 
lower  de7ioini7iation  tlian  thousands^  nor  a  higher  denomi- 
nation  thun  hundreds  of  thausayids. 

Hence,  if  a  number  contains  more  than  three  figures,  its 
cube  root  will  contain  more  than  one  :  if  it  contains  more 
than  six,  its  root  will  contain  more  than  two,  and  so  on ; 
every  additional  three  figures  giving  one  additional  figure  in 
the  root,  and  the  figures  which  remain  at  the  left  hand, 
although  less  than  three,  will  also  give  a  figure  in  the  root. 
This  law  explains  the  reason  for  pointing  off  into  periods  ol 
tliite  liguit>  each 


280 


CUBE    ROOT. 


300.  Let  us  now  see  how  the  cube  of  any  number,  as  1 6, 
is  formed.  Sixteen  is  composed  of  1  ten  and  6  units,  and 
may  ^e  written  10-f  6.  To  find  the  cube  of  16  or  of  10 -fO, 
we  must  multiply  the  number  by  itself  twice. 

To  do  this  we  place  the  number  thus  16=  10-f      6 

10+      6 
60+~36 


product  by  the  units 
product  by  the  tens 

Square  of  16 
Multiply  again  by  1 6 
product  by  the  units 
product  by  the  tens 

Cube  of  16 


100+     60 


100  + 


120  + 
10  + 


36 
6 


-    600+   720  +  216 
1000+1200+   360 


1000  +  1800+1080  +  216 

1.  By  examining  the  parts  of  this  number  it  is  seen  that 
the  first  part  1000  is  the  cube  of  the  tens  ;  that  is, 

10x10x10  =  1000. 

2.  The  second  part  1800  is   three  times  the  square  of  the 
tens  multiplied  by  the  units  ;  that  is, 

3  X  (10)2x6  =  3x100x6  =  1800. 

3.  The  third  part  1080  is  three  times  the  square  of  the  units 
multiplied  by  the  tens  ;  that  is, 

3x62x10  =  3x36x10=1080 

4.  The  fourth  part  is  the  cube  of  the  units  ;  that  is, 

62  =  6x6x6  =  216. 
1.   What  is  the  cube  root  of  the  numbei*  40961 


OPERATION. 

4  096(^6 
1 


Analysis. — Since  the  number 
contains  more  than  three  figures, 
we  know  that  the  root  will  eon- 
tain  at  least  units  and  tens. 

Separating  the  three  right- 
hand  figures  from  the  4,  we 
know  that  the  cube  of  the  tens 
will  be  found  in  the  4  ;  and  1  is  the  greatest  cube  in  4 


12x3  =  3)3  0     (9-8-7-6 
163  =  4  096. 


299.  What  is  the  cube  root  of  a  number  1  How  many  perfect  cubes 
are  there  between  1  and  1000  ! 

UOO.  Of  how  many  parts  is  the  cube  of  a  numbrr  c(iiii|M)«{>il  I  Wlial 
an-  ihfv '' 


OUBK    KOUl.  287  ' 

Hence,  we  place  the  root  1  ou  the  right,  and  this  is  the  tens  of 
the  required  root.  We  then  cube  1  and  subtract  the  result  from 
4,  and  to  the  remainder  we  brine  down  the  first  figure  0  of  the 
next  period. 

We  have  seen  that  the  second  part  of  the  cube  of  16,  viz  1800^ 
is  three  times  the  square  of  the  tens  multiplied  by  the  units  :  and 
hence,  it  can  have  no  significant  figure  of  a  less  denomination  tha» 
hundreds.  It  must,  therefore,  make  up  a  part  of  the  30  hundreds 
above.  But  this  30  hundreds  also  contains  all  the  hundreds 
which  come  from  the  3d  and  4th  parts  of  the  cube  of  16.  If  it 
were  not  so,  the  30  hundreds,  divided  by  three  times  the  square 
of  the  tens,  would  give  the  unit  figure  exactly. 

Forming  a  divisor  of  three  times  the  square  of  the  tens,  we  find 
the  quotient  to  be  ten;  but  this  we  know  to  be  too  large.  Placing 
9  in  the  root  and  cubing  19.  we  find  the  result  to  be  6859.  Then 
trying  8  we  find  the  cube  of  18  still  too  large;  but  when  we  take 
6  we  find  the  exact  number.  Hence,  the  cube  root  of  4096  is  16. 
301.  Hence,  to  find  the  cube  root  of  a  number, 
Rule. — 1.  Separate  the  given  number  into  periods  of  thret 
figures  eadi^  by  placing  a  dot  over  the  place  of  units,  a  second 
over  the  place  of  thousands,  and  so  on  over  each  third  figun 
to  the  left;  the  left  hand  period  will  often  contain  less  thai 
three  places  of  figures. 

II.  JVote  the  greatest  perfect  cube  in  the  first  period^  and 
set  its  root  on  the  right,  after  the  manner  of  a  quotient  in  di- 
vision. Subtract  the  cube  of  this  number  from  the  first  j^^iod^ 
and  to  the  remainder  bring  dovjn  the  first  figure  of  the  7iext 
period  for  a  dividend. 

III.  Take  three  times  the  square  of  the  root  just  found  fo^ 
a  trial  divisor,  and  see  how  often  it  is  contained  in  the  divi 
dend^  and  place  the  quotient  for  a  second  figure  of  the  root 
TJien  cube  the  fig\ires  of  the  root  thus  found,  and  if  theif 
cube  be  greater  than  the  first  two  periods  of  the  given  num- 
ber, diminish  the  last  figure,  but  if  it  be  less,  subtract  it 
from  the  first  ttco  periods,  a7id  to  the  remainder  bring  doicn 
the  first  figure  of  the  next  period  for  a  new  dividend. 

IV.  Take  three  times  the  square  of  the  whole  root  for  a 
second  trial  divisor,  and  find  a  third  figure  of  the  root. 
Cube  the  whole  root  thus  fou/nd  and  subtract  the  result  from 
the  first  three  periods  of  the  given  number  ivhen  it  is  less 
than  that  7iumher,  but  if  it  is  greater,  diniihish  the  figu^i 
of  the  root  ;  jnoceed  in  a  similar  way  for  all  the  periods. 


288  *^  ^ICUJiE    KOOT. 

EXAMPLES. 

1.   What  is  the  cube  root  of  99252b47  1 


99  252  847(463 
4^  =  64 


42x3  =  48)352       dividend. 
First  two  periods         -         -         -       99    252 
(46)^  =  46  X  46  X  46=  ^  336 

3  X  (46)2  =  6348  )     19168  2d  dividend. 
The  first  three  periods       -  99  252  847 

(463)3  =99  252  848 

Find  the  cube  roots  of  the  following  numbers  : 


1.  Of  389017? 

2.  Of  5735339  ? 

3.  Of  32461759? 


4.  Of  84604519? 

5.  Of  259694072  ] 

6.  Of  48228544  ? 


302.  To  extract  the  cube  root  of  a  decimal  fraction, 
Annex  ciphers  to  the  decimal^  if  necessary,  so  that  it 
shall  consist  of  3,  6,  9,  ^c,  places.  Then  put  the  first  point 
over  the  place  of  thousandths,  the  second  over  the  place  of 
millionths,  and  so  on  over  every  third  place  to  the  right  ; 
after  tvhich  extract  the  root  as  in  whole  numbers. 

Notes. — 1.  There  will  be  as  many  decimal  places  in  the  root 
as  there  are  periods  in  the  given  number. 

2.  The  same  rule  applies  when  the  given  number  is  compoaed 
of  a  whole  number  and  a  decimal. 

3.  If  in  extracting  the  root  of  a  number  there  is  a  remainder 
after  all  the  periods  have  been  brought  down,  periods  of  ciphers 
may  be  annexed  by  considering  them  as  decimals. 


EXAMPLES. 

Find  the  cube  roots  of  the  following  numbers  ; 

1.  Of  .157464? 

2.  Of  .870983875? 

3.  Of  12.977879  ? 

4.  Of  .7510894291 

5.  Of  .353393243  1 

6.  Of  3.408862625? 

301.  What  is  the  rule  for  extracting  the  cube  root? 

302.  How  do  you  extract  the  cube  root  of  a  decimal  fraction  1  How 
many  decimal  places  will  there  be  in  the  root  \  Will  the  same  rule 
apply  when  there  is  a  whole  number  and  a  decimaH  If  in  extracting 
the  root  of  any  f; umber  you  fmd  a  decimal,  how  do  you  proceed  I 


:bi=? 


APPLICATIomS.  z  "v>0 

%^ 

303.  To  extract  the  cube  root  of  a  o^mon  fraction. 

J.  B.educe  compound  fractiofis  to  simjdeov^es,  mixed  numr 
hers  to  improper  fractions,  and  then  reduce  the  fraction  to 
its  lowest  terms. 

II.  Extract  the  cube  root  of  the  numerator  and  denomi- 
lator  separately,  if  they  have  exact  roots ;  but  if  either  of 
hem  has  not  an  exact  root,  reduce  the  fraction  to  a  decimal, 
and  extract  the  root  as  %n  the  last  case, 

EXAMPLES. 

Find  the  cube  roots  oi"  the  following  fractions  : 
1    Of  ^^^-  2  I  4    Of  A  9 

2.  Of  31^1^?  5.  Of  f  ? 

3.  OfAVo^  I  6.  Off] 

APPLICATIONS. 

1.  What  must  be  the  length,  depth,  and  breadth  of  a  box, 
when  these  dimensions  are  all  equal  and  the  box  contains 
4913  cubic  feet  1 

2.  The  solidity  of  a  cubical  block  is  21952  cubic  yards  : 
what  is  the  length  of  each  side  %  What  is  the  area  of  the 
surface  ? 

3.  A  cellar  is  25  feet  long  20  feet  wide,  and  8^  feet  deep  : 
what  will  be  the  dimensions  of  another  cellar  of  equal  capactiy 
in  the  form  of  a  cube  1 

4.  What  will  be  the  length  of  one  side  of  a  cubical  granary 
that  shall  contain  2500  bushels  of  grain  % 

5.  How  many  small  cubes  of  2  inches  on  a  side  can  be 
sawed  out  of  a  cube  2  feet  on  a  side,  if  nothing  is  lost  in 
sawing  ? 

6.  What  will  be  the  side  of  a  cube  that  shall  be  equal  to 
the  contents  of  a  stick  of  timber  containing  1728  cubic  feet  ? 

7.  A  stick  of  timber  is  54  feet  long  and  2  feet  square  ; 
what  would  be  its  dimensions  if  it  had  the  form  of  a  cube  ] 

Notes. — 1.  Bodies  are  said  to  be  similar  when  their  like  part* 
are  propoitional. 

2.  It  is  touud  that  the  contents  of  similar  bodies  are  to  each 
other  as  the  cubes  of  their  like  dimensions. 


3()3.  How  do  you  extract  the  cube  root  of  a  vul{iar  fraction  \ 
11) 


21)0  AKITUMETICAL    PR0GKE6SI0N. 

3.   Ail    bodies  named   in  the  examples  are  supposed  to  be  simi 
lar. 

8.  If  a  sphere  of  4  feet  in  diameter  contains  33.5104  cubic 
feot,  what  will  be  the  contents  of  a  sphere  8  feet  in  diameter  1 

43     :     83     :     :     33.5104     :     Ans. 

9.  If  the  contents  of  a  sphere  14  inches  in  diameter  is 
1436.7584  cubic  inches,  what  will  be  the  diameter  of  a  sphere 
which  contains  11494-0672  cubic  inches? 

10.  If  a  ball  weighing  32  pounds  is  6  inches  in  diameter, 
what  will  be  the  diameter  of  a  ball  weighing  964  pounds  ? 

11.  If  a  haystack,  24  feet  in  height,  contains  8  tons  of  hay, 
what  will  be  the  height  of  a  similar  stack  that  shall  contain 
but  1  ton? 

ARITHMETICAL    PROGRESSION. 

304.  An  Arithmetical  Progression  is  a  series  of  numbers  in 
which  each  is  derived  from  the  preceding  one  by  the  addition 
or  subtraction  of  the  same  number. 

The  number  added  or  subtracted  is  called  the  common  dif- 
ference. 

305.  If  the  common  difference  is  added,  the  series  is  called 
an  increasing  series. 

Thus,  if  we  begin  with  2,  and  add  the  common  difference, 
3,  we  have 

2,     5,    8,    11,    14,    17,    20,    23,    &c., 

which  is  an  increasing  series. 

If  we  begin  with  23,  and  subtract  the  common  difference 
3,  we  have 

23,     20,     17,     14,     11,    8,    5,    &c., 
which  is  a  decreasing  series. 

304.  What  is  an  arithmetical  progression  T  What  is  the  number 
added  or  subtracted  called  1 

305.  When  the  common  diiference  is  added,  what  is  the  series  called  1 
What   is  it   called  when  the  common  difference  is  subtracted  1     What 

'  are  the  several   numbers  called  1     What  are  the   first  and  last  called  \ 
What  are  the  intermediate  uney  called  ! 


AlilTHMETIOAL    PliOGKILSSION.  291 

The  several  numbers  are  called  the  terms  of  the  progres- 
sion or  series  :  the  first  and  last  are  called  the  extremes,  and 
the  intermediate  terms  are  called  means. 

306.  In  every  arithmetical  progression  there  are  five 
parts  : 

1st,  the  first  term  ; 

2d,   the  last  term  ; 

3d,    the  common  difference  ; 

4th,  the  number  of  terms  ; 

5th,  the  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known  or  given,  the  remam- 
ing  ones  can  be  determined. 

CASE    I. 

307.  Knoiving  the  first  term,  the  common  difference,  and 
the  number  of  terms^  to  find  the  last  term. 

1.  The  first  term  is  3,  the  common  difference  2,  and  the 
number  of  terms  19  :  what  is  the  last  term? 

Analysis.  — By  considering  the  manner  in 
which  the  increasing  progression  is  formed,  we 
see  that  the  2d  term  is  obtained  by  adding  the 
common  difference  to  the  1st  term;  the  3d,  by  operation. 

adding  the  common  difference  to  the  2d  ;  the       18  No.      less    1 
4th,  by  addmg  the  common  difference  to  the         2  Com.  dif. 
3d,  and  so  on  ;  the  number  of  additions  being  1       o^ 
less  than  the  number  of  terms  found.  q    i  ^  + 

But  instead  of  making  the  additions,  we  may       ^     erm. 

multiply  the  common  difference  by  the  number       39   last  term, 
of  additions,  that  is,  by  1  less  than  the  number 
of  terms,    and  add  tlie   fii-st  term  to  the  pro- 
duct :  hence, 

Rule. — Multiply  the  common  difference  hy  1  less  than 
the  number  of  terms  ;  if  the  progression  is  increasing,  add 
tlie  'product  to  the  first  term  and  the  sum  will  be  the  last 
term  ;  if  it  is  decreasina^  subtract  the  product  frorri  t  le 
first  term  and  tJie  difference  will  be  the  last  term. 

306  How  many  parts  are  there  in  every  arithmetical  progression  ' 
What  are  they  1  How  many  parts  must  be  given  before  the  remaining 
ones  can  be  found  1 


292  AlilTUMETlCAL    PKOGliKSSION. 


EXAMPLES. 


1.  A  man  bought  50  yards  of  cloth,  for  which  he  was  to 
pay  6  cents  for  the  1st  yard,  9  cents  for  the  2d,  12  cents  for 
the  3d,  and  so  on  increasing  by  the  common  difference  3  : 
how  much  did  he  pay  for  the  last  yard  1 

2.  A  man  puts  out  $100  at  simple  interest,  at  7  per  cent : 
at  the  end  of  the.  1st  year  it  will  have  increased  to  $107,  at 
the  end  of  the  2d  year  to  $114,  and  so  on,  increasing  $7 
each  year  :  what  will  be  the  amount  at  the  end  of  16  years  1 

3.  What  is  "the  40th  term  of  an  arithmetical  progression  of 
which  the  first  term  is  1,  and  the  common  difference  1  i 

4.  What  is  the  30th  term  of  a  descending  progression  of 
which  the  first  term  is  60,  and  the  common  difierence  2  ? 

5.  A  person  had  35  children  and  grandchildren,  and  it  so 
happened  that  the  difierence  of  their  ages  was  18  months, 
and  the  age  of  the  eldest  was  60  years  :  how  old  was  the 
youngest  ] 

CASE    II. 

308.  Knowing  the  two  extremes  and  the  number  of  terms^ 
to  Jind  the  common  difference. 

1.  The  extremes  of  an  arithmetical  progression  are  8  and 
104,  and  the  number  of  terms  25  :  what  is  the  common  dii- 
ference  ?  4^ 

Analysis. — Since  the  common  difference 
multiplied  by  1  less  than  the  number  of  operation. 

^erms  gives  a  product  equal  to  the  differ-  1.04 

eiice  of  the  extremes,  if  we  divide  the  dif-  g 

ference  of  the  extremes  by  1  less  than  the         —: oTVdr'TT 

number  of  terms,  the  quotient  will  be  the         '^^—  1  =i/i4)  Jo(4. 
common  difference :  hence, 

Rule. —  Subtract  the  less  extreme  from,  the  grealer  and 
divide  the  remainder  by  1  Less  than  the  number  of  tertns  ; 
the  quotient  will  be  the  common  difference. 


307.  When  you  know  the  first  term,  the  cuniiiioti  difierence,  and  the 
atimber  of  terms,  how  do  you  find  the  last,  term  ! 

308.  When  you  know  the  extremes  and  the  number  of  tenus,  how  do 
you  find  the  common  difference  \ 


AJirmMi.  rioAL  pikkjrkssion.  29'J 

EXA?.irLES. 

1.  A  man  has  8  sons,  the  youngest  is  4  years  old  and  the 
eldest  32  :  their  ages  increase  in  arithmetical  progression : 
what  is  the  common  diflerence  of  their  ages  ? 

2.  A  man  is  to  travel  from  New  York  to  a  certain  place  in 
12  days ;  to  go  3  miles  the  first  day,  increasing  every  day  by 
the  same  number  of  miles  ;  the  last  day's  journey  is  58  miles: 
required  the  daily  increase. 

3.  A  man  hired  a  workman  for  a  month  of  26  working 
days,  and  agreed  to  pay  him  50  cents  for  the  first  day,  with 
a  uniform  darily  increase;  on  the  last  day  he  paid  $1,50: 
what  was  the  daily  increase  ? 

CASE  III. 

309.  To  find  the  sum  of  the  terms  of  an  arithmetical 
progression. 

1.  What  is  the  sum  of  the  series  whose  first  term  is  3, 
common  diflerence  2,  and  last  term  19  ? 

Given  series    -   3+    5-f-    7+    9-1-114-134-14+17  +  19=    99 

Sarae;  order  )  . 

of  terms  in-[   19+17+15+13+11+    9+    V+    5+    3=    99 

inverted.        J 

Sum  of  both.    22     22     22     22     22     22     22     22     22  =  198 

Analysis. — The  two  series  are  the  sarae  ;  hence,  their  sum  is 
equal  to  twice  the  given  series.  But  tlieir  sum  is  equal  to  the 
sum  of  the  two  extremes  3  and  19  taken  as  many  times  as  there 
are  terms  :  and  the  given  series  is  equal  to  half  this  sum,  or  to 
the  sum  of  the  extremes  multiplied  by  half  the  number  of  terms. 

Rule. — Add  the  extremes  together  and  multiply  their 
sutn  by  Jialf  the  number  of  terms  ;  the  'product  will  be  the 
sum  of  tlie  series. 

EXAMPLES. 

1.  The  extremes  are  2  and  100,  and  the  number  of  terms 
22  :  what  is  the  sum  of  the  series  1 

OPERATION. 

2   1st  term, 
100  last  terra. 


Analysis. — We  first   add 
together  the  two  extremes 


and  then  multiply  by  half  102  sum  of  extremes. 

the  number  of  terms.  1  ]    half  the  number  of  terms. 

1122  sum  of  series. 

1}0U.   How  do  you  find  the  sum  of  the  tcrmbl 


2t5  I  GLOMLTKICAL    J'KOOKESSION. 

2.  How  many  strokes  does  the  hammer  of  a  clock  strike  in 
12  hours? 

3.  The  first  term  of  a  series  is  2,  the  common  diflerence  4, 
and  the  number  of  terms  9  :  what  is  the  last  term  and  sum  of 
the  series  ? 

4.  James,  a  smart  chap,  having  learned  arithmetical  pro- 
gression, told  his  father  that  he  would  chop  a  load  of  wood  of 
1 5  logs,  at  2  cents  the  first  log,  with  a  regular  increase  of 
I  cent  for  each  additional  log :  how  much  did  James  receive 
for  chopping  the  wood  ? 

5.  An  invalid  wishes  to  gain  strength  by  regular  and  in- 
creasing exercise  ;  his  physician  assures  him  that  he  can 
walk  1  mile  the  first  day,  and  increase  the  distance  half  a 
mile  for  each  of  the  24  following  days  :  how  far  will  he 
walk? 

6.  If  100  eggs  are  placed  in  a  right  line,  exactly  one  yard 
from  each  other,  and  the  first  one  yard  from  a  basket  :  what 
distance  will  a  man  travel  who  gathers  them  up  singly,  and 
places  them  in  the  basket  ] 


GEOMETRICAL    PROGRESSION. 

310.  A  Geometrical  Progression  is  a  series  of  terms, 
each  of  which  is  derived  from  the  preceding  one,  by  multi- 
plying it  by  a  constant  number.  The  constant  multipher  is 
called  the  ratio  of  the  progression. 

311.  If  the  ratio  h  greater  than  1,  each  term  is  greater 
than  the  preceding  one,  and  the  series  is  said  to  be  in- 
creasing. 


310.  What  is  a  geometrical  progression!  What  is  the  constant 
multipher  called  1 

311.  If  the  ratio  is  greater  than  1,  how  do  the  terms  compare  with 
each  other  1  What  is  the  series  then  called  1  If  the  ratio  is  less 
than  1,  how  do  they  compare?  WTiat  is  the  series  then  called  ?  What 
are  the  several  numbers  called  1  What  are  the  first  and  last  called  ? 
What  are  the  intermediate  ones  called  1 

312.  How  many  parts  are  there  in  every  geometrical  progression  1 
What  are  they  1  How  n.'any  must  be  known  before  the  others  can  be 
fuund  ** 


1,.    2, 

4, 

8,   16,  32,  &c.- 

52,   16, 

8, 

4,    2,      1,    &c.- 

GEOMEITJICAL    PROGRESSION.  20*i 

If  the  ratio  is  less  than  1,  each  term  is  less  than  the 
preceding  one,  and  the  series  is  said  to  be  decreasing ; 
thus, 

-ratio  2 — increasing  series  : 
-ratio  ^^-decreasing  series. 

The  several  numbers  are  called  terms  of  the  progression. 
The  first  and  last  are  called  the  extremes,  and  the  intermedi- 
ate terms  are  called  means. 

312.  In  every  Geometrical,  as  well  as  in  every  Arithmeti- 
'^al  Progression,  there  are  five  parts  : 

1st,    the  first  term  ; 
2d^  the  last  term  ; 
3d,    the  common  ratio  ; 
4th,  the  number  of  terms  ; 
5th.  the  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known,  or  given,  the  re- 
training ones  can  be  determinied. 

CASE    I. 

313.  Having  given  the  first  term,  the  raXio,  and  the 
nwinber  of  terms,  to  find  tlie  last  term. 

1.  The  first  term  is  3  and  the  ratio  2  :  what  is  the  6th 
term  1 

Analysis, — The    se-  opkration. 

oond  term  is  formed  by       2  X  2  X  2  X  2  X  2  =  2^  =r  32 
multiplying     the      first  3   ^^^  ^^ 

term   by  the  ratio  ;  the  

third  term  by  multiply-  Ans.     96 

ing  the  second  term   by 

the  ratio,  and  so  on ;  the  number  of  multiplicators  being    I  /«5i 

ihan  the  number  of  terms :  thus, 

3  =  3  1st   term, 

3x2  =  6  2d    term, 

3x2x2  =  3x22=12   3d    term, 
15x2x2x2  =  3x23  —  24  4th  term,  &c. 


2D(J  GEOMETRICAL    FKOGKESSION. 

Therefore,  the  last  term  is  equal  to  the  f/st  ternCmulti- 
plied  by  the  ratio  raised  to  a  power  1  less  than  the  numbet 
of  tenns. 

Rule. — Raise  the  ratio  to  a  power  whose  ex-ponent  is  1 
less  than  the  number  of  terms,  and  then  multiply  this  powei 
by  the  first  term. 

EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  192  ;  the 
ratio  J,  and  the  number  of  terms  7  :  what  is  the  last  term '? 

Note. — The  6th   power  of  the  ratio,  (i,)  is  operation. 

^^  and  this  multiplied   by  the  first   term  192,  (i)'^=:Jj 

gives  the  last  term  3.  192  X  ^~ 3. 

2.  A  man  purchased  1 2  pears  ;  he  wa  to  pay  1  farthing 
for  the  1st,  2  farthnigs  for  the  2d,  4  for  ihe  3d,  and  so  on, 
doubhng  each  time  :   what  did  he  pay  for  the  last  ? 

3.  The  first  term  of  a  decreasing  progression  is  1024,  the 
ratio  ^  :  what  is  the  9th  term  1 

4.  The  first  term  of  an  increasing  progression  is  4,  and  the 
common  difierence  3  :  what  is  the  10th  term  ] 

0.  A  gentleman  dying  left  nine  sons,  and  bequeathed  his 
estate  in  the  IblloM'ing  manner  :  to  his  executors  $50  ;  his 
youngest  son  to  have  twice  as  much  as  the  executors,  and 
each  son  to  have  double  the  amount  of  the  son  next  younger  : 
what  was  the  eldest  son's  portion  \ 

6.  A  man  bought  12  yards  of  cloth,  giving  3  cents  ibr  the 
1st  yard,  6  for  the  2d,  12  for  the  3d,  &o.  :  what  did  he  pay 
for  the  last  yard  V 

CASE  n. 

314.  Knovnng  the  two  extremes  and  the  ratio,  to  find 
the  sum  of  the  terms. 

1.  What  is  the  sum  ol  the  terms,  in  the  progression,  1,  4, 
16,  64? 

313.  Knowing  the  first  term,  the  r^tio,  and  the  number  of  terms,  how 
do  you  find  the  last  term "? 

314.  Knowing  the  two  extremes  and  the  ratio,  how  do  you  fmd  the 
sum  of  tlie  terms  ? 


GEOMLTKICAL    PKoGKESSlO^.  297 

Analysis. — If  we  multiply  the  terms  of  the  progression  by  the 
ratio  4.  we  have  a  second  pro- 
gression,. 4,  16,  64,  256,  which  operation. 
is  4  times  as  great  as  the  first.           4+16  +  64  +  256=        4  times. 
If  from  this  we  subtract  the    1+4  +  16  +  64  =  once, 
first,  the  remainder,   256  —  1,                                 Oof  — 1  =Tf 
will  be  3  times   as  great   as                                                 '  ' 
the  first;  and  if  the  remain-          256—1     255 

der  b«  divided  by  3,  tlie  quo-  ~§ =  — :=85  sum. 

tieut  will   be  the  sum  of  the 

terms  of  the  first  progression.  But  256  is  the  product  of  the  last 
term  of  the  given  progression  multiplied  by  the  ratio,  1  is  the  first 
term,  and  the  divisor  3  is  1  less  ihan  the  ratio  :  hence. 

Rule. — Multiply  the  last  term  by  the  ratio  ;  take  the  dif- 
ference between  the  jn-oduct  and  the  first  term  and  divide 
the  remainder  by  the  difference  between  1  and  the  ratio. 

Note.-— When  the  progression  is  increasing^  the  first  term  is 
subtracted  from  the  product  of  the  last  term  by  the  ratio,  and  the 
divisor  is  found  by  subtracting  1  from  the  ratio.  When  the  pro- 
gression is  decreasing^  the   product  of  the  last  term  by  the  ratio  is 


EXAMPLES. 

1 .  The  first  term  of  a  progression  is  2,  the  ratio  3,  and  the 
last  term  4374  :   what  is  the  sum  of  the  terms  ? 

2.  The  first  term  of  a  progression  is  128,  the  ratio  i,  and 
the  last  term  2  :  what  is  the  sum  of  the  terms  1 

?>.  The  first  term  \&  3,  the  ratio  2,  and  the  last  term  192  : 
what  is  the  sum  of  the  series  '? 

4.  A  gentleman  gave  his  daughter  in  marriage  on  New 
Year's  day,  and  gave  her  husband  \s.  towards  her  portion, 
and  was  to  double  it  on  the  first  day  of  every  month  during 
the  year  :  what  was  her  portion  % 

5.  A  man  bought  10  bushels  of  wheat  on  the  condition  that 
he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  2d,  9  for 
the  3d,  and  so  on  to  the  last :  what  did  he  pay  for  the  last 
bushel,  and  for  the  1 0  bushels  1 

6.  A  man  has  6  children  :  to  the  1st  he  gives  $150,  to  the 
2d  $300,  to  the  3d  $600,  and  so  on,  to  each  twice  as  much 
as  the  last :  how  much  did  the  eldest  receive,  and  what  was 
Die  amount  received  by  them  all  ? 


21)8  PKOMISCUOUS    QUESTIONS. 


PROMISCUOUS    EXAMPLES. 


1.  A  merchant  bought  13  packages  of  goods,  lor  which  he  paid 
$326  :  what  will  39  packages  cost  at  the  same  rate? 

2.  How  many  bushels  of  oats  at  62-J  cents  a  bushel  will  pay 
for  4250  feet  of  lumber  at  $7,50  per  thousand? 

3.  Bought  Ihhd.  of  sugar  which  weighed  as  follows  :  the  1st 
'^cwi.  \qr.  18lb.,  the  2d  6cwt.  10/6.  :  what  did  it  cost  at  7  cents  per 
pound? 

4.  How  many  hours  between  the  4th  of  Sept.,  1854,  at  3  P.M., 
and  the  20th  day  of  April,  1855,  at  10  A.M.  ? 

5.  If  I  of  a  gallon  of  wine  cost  -f  of  a  dollar,  what  will  f  of  a 
hogshead  cost  ? 

6.  What  number  is  thai  which  being  multiplied  by  |  will  pro- 
duce ^? 

7.  A  tailor  had  a  piece  of  cloth  containing  24j  yards,  from  which 
he  cut  6f  yards  :  how  much  was  there  left  ? 

8.  From  |  of  ^  take  J  of  ^^ 

^12  ^5 

9.  What  is  the  difference  between  3|  +  7f  and  4  +  2^  ? 

10.  There  was  a  company  of  soldiers,  of  whom  \  were  on  guard, 
^  preparing  dinner,  and  the  remainder,  85  men,  were  drilling : 
how  many  were  there  in  the  company  ? 

11.  The  sum  of  two  numbers  is  425,  and  their  difference'  1.625 : 
what  are  the  numbers  ? 

12.  The  sum  of  two  numbers  is  ^.  and  their  difference  i|^:  what 
are  the  numbers  ? 

13.  The  product  of  two  numbers  is  2.26,  and  one  of  the  numbers 
is  .25  :  what  is  the  other  ? 

14.  If  the  divisor  of  a  certain  number  be  6.66f,  and  the  quo- 
tient I,  what  will  be  the  dividend  ? 

15.  A  person  dying,  divided  his  property  between  his  widow  and 
his  four  sons;  to  his  widow  he  gave  $1780,  and  to  each  of  his 
sons  $1250  ;  he  had  been  25^  years  in.  business,  and  had  cleared 
on  an  average  $126  dollars  a  year:  how  much  had  he  when  he 
began  business  ? 

16.  A  besieged  garrison  consisting  of  360  men  was  provisioned 
for  6  months,  but  hearing  of  no  relief  at  the  end  of  five  months, 
dismissed  so  many  of  the  garrison,  that  the  remaining  provision 
lasted  5  months  :  how  many  men  were  sent  away  ? 

17.  Two  persons,  A  and  B  are  indebted  to  C  ;  A  owes  $2173, 
which  is  the  least  debt,  and  the  difference  of  the  debts  is  Vo71  : 
what  is  the  amount  of  their  indebtedness  ? 

18.  What  number  added  to  the  43d  part  of  4429,  will  make  the 
8uni  240  ? 

20 


PliOMlSCUUUS    QUEBTIONS.  209 

19.  How  many  planks  15  feet  long,  and  15  inches  wide,  will 
floor  a  barn  60^  feet  long,  and  33^  feet  wide? 

20.  A  person  owned  -|  of  a  mine,  and  sold  |  of  his  interest  for 
$1710  :  what  was  the  value  of  the  en  lire  mine? 

'21.  A  room  30  feet  long,  and  18  feet  wide,  is  to  be  covered  with 
painted  cloth  f  of  a  yard  wide  :  how  many  yards  will  cover  it  ? 

22.  A,  B  and  C  trade  together  and  gain  $120,  which  is  to  be 
shared  according  to  each  one's  stock;  A  put  in  $140,  B  $300,  and 
C  $160  :  what  is  each  man's  share? 

23.  A  can  do  a  piece  of  work  in  12  days,  and  B  can  do  the  same 
work  in  18  days  :  how  long  wil]  it  take  both,  if  they  work  togethor  ? 

24.  If  a  barrel  of  flour  will  iast  one  family  7^  months,  a  .second 
family  9  months,  and  a  third  11^  months,  how  long  will  it  last  the 
three  families  together  ? 

25.  Suppose  I  have  ^|  of  a  ship  worth  $1200  ;  what  part  have 
I  left  after  selling  ^  of  ^  of  my  share,  and  what  is  it  worth  ? 

26.  What  number  is  that  which  being  multiplied  by  |  of  |  of 
l^,  tho  product  will  be  1  ? 

27.  Divide  $420  between  three  persons,  so  that  the  second  shall 
have  ^  as  much  as  the  first,  and  the  third  ^  as  much  as  the  other  two  ? 

28.  What  is  the  difference  between  twice  five  and  fifty,  and 
twice  fifty-five  ? 

29.  What  number  is  that  which  being  multiplied  by  three- 
thousandths,  the  product  will  be  2637  ? 

30.  What  is  the  difference  between  half  a  dozen  dozens  and  six 
dozen  dozens  ? 

31.  The  slow  or  parade  step  is  70  paces  per  minute,  at  28  inches 
each  pace :  how  fast  is  that  per  hour  ? 

32.  A  lady  being  asked  her  age.  and  not  wishing  to  give  a  direct 
answer,  said,  *•  I  have  9  children,  and  three  years  elapsed  between 
the  birth  of  each  of  them ;  the  eldest  was  born  w/ien  I  was  1 9 
years  old.  and  the  youngest  is  now  exactly  19  :"  what  was  her  age  ? 

33.  A  wall  of  700  yards  in  length  was  to  be  buxlt  in  29  days  : 
12  men  were  employed  on  it  for  11  days,  and  only  completed  220 
yards  :  how^  many  men  must  be  added  to  complete  the  wall  in  the 
required  time  ? 

34.  Divide  $10429,50  between  three  persons,  so  that  as  often 
as  one  gets  $4,  the  second  will  get  $6,  and  the  third  $7. 

35.  A  gentleman  whose  annual  income  is  £1500,  spends  20 
guineas  a  week  :  does  he  save,  or  run  in  debt,  and  how  much  ? 

36.  A  farmer  exchanged  70  bushels  of  rye,  at  $0,92  per  bushel, 
for  40  bushels  of  wheat,  at  $1,37-^  a  bushel,  and  received  the 
balance  in  oats,  at  $0,40  per  bushel :  how  many  bushels  of  oats 
did  he  receive  ? 

37.  In  a  certain  orchard  -^  of  the  trees  bear  apples,  ^  of  them 
Lear  peaches,  ^  of  them  plums,  120  of  them  cherries,  and  80  of 
Ihem  pears  :  how  many  trees  are  there  in  the  orcliard  ? 


000  PKOMlSClIOLife    QUESTIONS. 

38.  A  person  being  asked  the  time,  said,  the  time  past  noon 
IS  equal  to  \  of  the  time  past  midnight :  what  was  the  hour? 

39.  If  20  men  can  perform  a  piece  of  work  in  12  days,  how 
many  men  will  accomplish  thrice  as  much  in  one-fifth  of  the  time  ? 

40.  How  many  stones  2  feet  long.  1  foot  wide,  and  6  inches 
thick,  will  build  a  wall  12  yards  long,  2  yards  high,  and  4  feet 
thick  ? 

41.  Four  persons  traded  together  on  a  capital  of  $6000.  of 
which  A  put  in  ^,  B  put  in  ^.  C  put  in  J,  and  D  the  rest ;  at  the 
end  of  4  years  they  had  gained  $4728  :  what  was  each  one's 
share  of  the  gain  ? 

42.  A  cistern  containing  60  gallons  of  water  hah  three  unequal 
cocks  for  discharging  it ;  the  largest  will  empty  it  in  one  hour,  the 
second  in  two  hours,  and  the  third  in  three  hours  :  in  what  time 
will  the  cistern  be  emptied  if  they  run  together  ? 

43.  A  man  bought  f  of  the  capital  of  a  cotton  factory  at  par; 
he  retained  |-  of  his  purchase,  and  sold  the  balance  for  $5000, 
which  was  15  per  cent  advance  on  the  cost ;  what  was  the  whole 
capital  of  the  factory? 

44.  Bought  a  cow  for  $30  cash,  and  sold  her  for  $35  at  a  credit 
of  8  months :   reckoning  the  interest  at  6  per  cent,  how  much  did 

1  gain  ? 

45.  If,  when  I  sell  cloth  for  85.  9c?.  per  yard,  I  gain  12  per  cent, 
what  will  be  the  gain  when  it  is  sold  for  IO5.  6d.  per  yard  ? 

46.  How  much  stock  at  par  value  can  be  purchased  for  $8500, 
at  8-J  per  cent  premium,  \  per  cent  being  paid   to  the  broker  ? 

47.  Twelve  workmen,  working  12  hours  a  day,  have  made,  in 
12  days,  12  pieces  of  cloth,  each  piece  75  yards  long  :  how  many 
pieces  of  the  same  stuff  would  have  been  made,  each  piece  25 
yards  long,  if  there  had  been  7  more  workmen  ? 

48.  A  person  was  born  on  the  1st  day  of  Oct.,  1801,  at  6  o'clock 
in  the  morning:  what  w^as  his  age  on  the  21st  of  Sept.,  1854,  at 
half-past  4  in  the  afternoon? 

49.  A,  can  do  a  piece  of  work  alone  in  10  days,  and  B  in  13 
days :  in  what  time  can  they  do  it  if  they  work  together  ? 

50.  A  man  went  to  sea  at  17  years  of  age  ;  8  years  after  he 
had  a  son  born,  who  lived  46  years,  and  died  before  his  father  ; 
after  which  the  father  lived  twice  twenty  years  and  died  :  what 
was  the  age  of  the'  father  ? 

51.  How  many  bricks,  8  inches  long  and  4  inches  wide,  will 
pave  a  yard  that  is  100  feet  by  50  feet  ? 

52.  If  a  house  is  50  feet  wide,  and  the  post  which  supports  the 
ridge  pole  is  12  feet  high,  what  will  be  the  length  of  the  rafters? 

53.  A  man  had  12  sons,  the  youngest  was  3  years  old  and  the 
eldest  58,  and  their  ages  increased  in  Arithmetical  progression 
what  wai;  tlic  comnioii  diflerence  of  their  ages  ? 


PKOlIlBCUOUtJ    tiUlLSTlUNS.  iiUl 

54.  If  a  quantity  of  provisions  serves  1500  men  12  weeks,  tit 
tlie  rate  of  20  ounces  a  day  for  each  man,  how  many  men  will  the 
same  provisions  maintain  for  20  weeks,  at  the  rate  of  8  ounces  a 
day  for  each  man? 

55.  A  man  bought  10  bushels  of  wheat,  on  the  condition  that 
he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  2d,  9  for  the  3d, 
and  so  on  to  the  last :  what  did  he  pay  for  the  last  bushel,  and  for 
the  10  bushels? 

56.  There  is  a  mixture  made  of  wheat  at  45,  per  bushel,  rye  at 
3s.,  barley  at  25,,  with  12  bushels  of  oats  at  18f/.  per  bushel  :  how 
much  must  be  taken  of  each  sort  to  make  the  mixture  worth  35. 
6d.  per  bushel  : 

57.  What  length  must  be  cut  ofT  a  board  8^  inches  broad  to 
contain  a  square  foot  ? 

58.  What  is  the  difference  between  the  interest  of  $2500  for  4 
years  9  mo.  at  6  per  cent,  and  half  that  sura,  for  twice  the  time, 
at  half  the  same  rate  per  cent  ? 

59.  A  person  lent  a  certain  sum  at  4  per  cent  per  annum ;  had 
this  remained  at  interest  3  years,  he  would  have  received  for  prin- 
cipal and  interest  $9676,80  :  what  was  the  principal  ? 

60.  If  1  pound  of  tea  be  equal  in  value  to  50  oranges,  and  70 
oranges  be  worth  84  lemons,  what  is  the  value  of  a  pound  of  tea, 
when  a  lemon  is  worth  2  cents  ? 

til.  A  person  bought  160  oranges  at  2  for  a  penny,  and  180 
more  at  3  lor  a  penny  ;  after  which  he  sold  them  out  at  the  rate 
of  5  for  2  pence :  did  he  make  or  lose,  and  how  much  ? 

62.  A  snail  in  getting  up  a  pole  20  feet  high,  was  observed  to 
climb  up  8  feet  every  day,  but  to  descend  4  feet  every  night  :  in 
what  time  did  he  reach  the  top  of  the  pole  ? 

63.  A  ship  has  a  leak  by  which  it  would  fill  and  sink  in  15 
hours,  but  by  means  of  a  pump  it  could  be  emptied,  if  full,  in 
16  hours.  Now,  if  the  pump  is  -worked  from  the  time  the  leak 
begins,  how  long  before  the  ship  will  sink  ? 

64.  A  and  B  can  perform  a  certain  piece  of  work  in  6  days.  B 
and  C  in  7  days,  and  A  and  C  in  14  days  :  in  what  time  would 
each  do  it  alone? 

65.  Divide  $500  among  4  persons,  so  that  when  A  has  -J  dollar, 
B  shall  have  f  C  \,  and  D  |, 

66.  A  man  purchased  a  building  lot  containing  3600  square 
feet,  at  the  cost  of  $1.50  per  foot,  on  which  he  built  a  store  ar  an 
expense  of  $3000.  He  paid  yearly  $180.66  for  repairs  and  taxes: 
what  annual  rent  must  he  receive  to  obtain  10  per  cent,  on  the 
cc«t  ? 

67.  A's  note  of  $7851,04  was  .dated  Sept,  5th,  1837,  on  which 
were  endorsed    the   following   payments,  viz.:  Nov.  13th,  1839, 
$410,98;  May  lOtli,  1840,  $152  :  what  wus  due  March  Itt.,  1841 
ihe  inteic:t  hciu^^.  (>  per  cent  ? 


:30'2  I'KOMISCUOUS    QUEBTlOiS'S. 

68.  A  house  is  40  feet  from  tlie  ground  to  the  eaves,  and  it  is 
required  to  find  the  length  of  a  ladder  which  will  reach  the  eaves, 
supposing  the  foot  of  the  ladder  cannot  be  placed  nearer  to  the 
house  than  30  feet  ? 

69.  Sound  travels  about  1142  feet  in  a  second;  now,  if  the 
flash  of  a  cannon  be  seen  at  the  moment  it  is  fired,  and  the  report 
heard  45  seconds  after,  what  distance  would  the  observer  be  from 
the  gun  ? 

70.  A  person  dying,  worth  $5460,  left  a  wife  and  2  children,  a 
son  and  daughter,  absent  in  a  tbreign  country.  He  directed  that 
if  his  son  returned,  the  mother  should  have  one-third  of  the  estate, 
and  the  son  the  remainder  ;  but  if  the  daughter  returned,  she 
should  have  one-third,  and  the  mother  the  remainder,  Now  it  so 
happened  that  they  both  returned  :  how  must  the  estate  be  divided 
to  fulfill  the  father's  intentions  ? 

71.  Two  persons  depart  from  the  same  place,  one  travels  32, 
and  the  other  36  miles  a  day  :  if  they  travel  in  the  same  direction, 
how  far  will  they  be  apart  at  the  end  of  19  days,  and  how  far  if 
they  travel  in  contrary  directions? 

72.  In  what  time  will  $2377,50  amount  to  $2852,42,  at  4  per 
cent,  per  annum  ? 

73.  What  is  the  height  of  a  wall,  which  is  14^  yards  in  length, 
and  y'^  of  a  yard  in  thicknes.^,  and  which  has  cost  $406,  it  having 
been  paid  for  at  the  rate  of  $10  per  cubic  yard  ? 

74.  What  will  be  the  duty  on  22o  bags  of  coffee,  each  weighing 
gross  1 60  lbs.,  invoiced  at  6  cents  per  lb.  :  2  per  cent,  being  the 
legal  rate  of  tare,  and  20  per  cent,  the  duty  ? 

75.  Three  persons  purcliase  a  piece  of  property  for  $9202  ;  the 
first  gave  a  certain  sum  ;  the  second  three  times  as  much  ;  and 
the  third  one  and  a  half  times  as  much  as  the  other  two  :  what 
did  each  pay  ? 

76.  A  reservoir  of  water  has  two  cocks  to  supply  it.  The  first 
would  fill  it  in  40  minutes,  and  the  second  in  50.  It  has  likewise 
a  discharging  cock,  by  which  it  may  be  emptied  when  full  in  25 
minutes.  Now,  if  all  the  cocks  are  opened  at  once,  and  the  water 
runs  uniformly  as  we  have  supposed,  how  long  before  the  cisterr 
will  be  filled  ? 

77.  A  traveller  leaves  New  Haven  at  8  o'clock  on  Monday 
morning,  and  walks  towards  Albany  at  the  rate  of  3  miles  an 
hour  :  another  traveller  sets  out  from  Albany  at  4  o'clock  on  the 
same  evening,  and  walks  towards  New  Haven  at  the  rate  of  4 
miles  an  hour;  now,  supposing  the  distance  to  be  130  miles, 
where  on  the  road  will  they  meet  ? 


MENSUKATION. 


30:J 


MENSURATION. 

315.  A  triangle  is  a  portion  of  a  plane 
bounded  by  three  straight  lines.  BC  is 
called  the  base  ;  and  AD,  perpendicular  to 
BC,  the  altitude. 

316.  To  find  the  area  of  a  triangle 
Tke  area  or  contents  of  a  triangle  is  equal 

to  half  the  product  of  its  base  by  its  altitude 
{Bk.  IV.  Prop.  VI).* 

EXAMPLES. 

1.  The  base,  BC,  of  a  triangle  is  40  yards,  and  the  perpendicu 
lar,  AD,  20  yards  :  what  is  the  area? 

2.  In  a  triangular  field  the  base  is  40  chains,  and  the  perpendi- 
cular 15  chains  :  how  much  does  it  contain  ?     (Art.  1 10.) 

3.  There  is  a  triangular  field,  of  which  the  base  is  35  rods  and 
the  perpendicular  26  rods :  what  are  its  contents  ? 


317.   A  square  is  a  figure  having  four  equal  sides, 
ojid  all  its  angles  riuht  angles. 


318.  A  rectangle  is  a  four-sided  figure  like  a 
square,  in  which  the  sides  are  perpendicular  to  each 
other,  but  the  adjacent  sides  are  not  equal. 

319.  A  parallelogram  is  a  four-sided  figure 
which  has  its  opposite  sides  equal  and  parallel,  but 
it»  angles  not  right  angles.  The  line  DE,  perpendi- 
cular to  the  base,  is  called  the  altitude. 

320.  To  find  the  area  of  a  square,  rectangle,  or  parallelogram, 
Multiply  the  base  by  the  perpendicular  height,  and  the  product 

will  be  the  area.     {Book  IV.  Prop.  V). 

EXAMPLES. 

1.  What  is  the  area  of  a  square  field  of  which  the  sides  are 
each  33.08  chains  ? 

2.  What  is  the   area  of  a  square  piece  of  land  of  which  the 
sides  are  27  chains  ? 

3.  What  is  the  area  of  a  square  piece  of  land  of  which  the  sides 
are  25  rods  each  ? 


D 


h 


E 


All  the  reference*:  are  to  Davies'  LefjenJre. 


304  MENSURATION. 

4.  What  are  the  contents  of  a  rectangular  field,  the  length  oi 
which  is  40  rods  and  the  breadth  20  rods  ? 

5.  What  are  the  contents  of  a  field  40  rods  square  ? 

6.  What  are  the  contents  of  a  rectangular  field  15  chains  long 
and  5  chains  broad  ? 

7.  What  are  the  contents  of  a  field   27  chains  long  and  9  rods 
broad  ? 

8.  The  base  of  a  parallelogram  is  271  yards,  and  the  perpendi 
cular  height  360  feet :  what  is  the  area  ? 

321.   A  trapezoid  is  a  four-sided  figure  -0_ 

ABCD,  having  two  of  its  sides,  AB,  DC, 
parallel.     The  perpendicular  CE  is  called 


/ 


the  altitude.  A  E  B 

322.  To  find  the  area  of  a  trapezoid. 

Multiply  the  sum  of  the  two  parallel  sides  by  the  altitude,  and 
divide  the  product  by  2,  the  quotient  will  be  the  area.  (Bk.  IV. 
Prop.  VII). 

EXAMPLES. 

1 .  Required  the  area  of  the  trapezoid  ABCD,  having  given 
AB=321.51/i^.,     DC-=214.24/^,     and  CE=171.16/^ 

2.  What  is  the  area  of  a  trapezoid,  the  parallel  sides  of  which 
are  12.41  and  8.22  chains,  and  the  perpendicular  distance  between 
them  5.15  chains  ? 

3.  Required  the  area  of  a  trapezoid  whose  parallel  sides  are  25 
feet  6  inches,  and  18  feet  9  inches,  and  the  perpendicular  distance 
between  them  10  feet  and  5  inches. 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides  are 
20.5  and  12.25,  and  the  perpendicular  distance  between  them 
10.75  yards. 

5.  What  is  the  area  of  a  trapezoid  whose  parallel  sides  are  7.50 
chains,  and  12.2^chains,  and  the  perpendicular  height  15.40  chains  ? 

6.  What  are  the  contents  when  the  parallel  sides  are  20  and  32 
chains,  and  the  perpendicular  distance  between  them  26  chains  ? 

323.  A  circle  is  a  portion  of  a  plane 
bounded  by  a  curved  line,  called  the  circum- 
ference. Every  point  of  the  circumference  is 
equally  distant  from  a  certain  point  within 
called  the  centre :  thus,  C  is  the  centre,  and 
any  line,  as  ACB,  passing  through  the  centre, 
is  called  a  diameter. 

If  the  diameter  of  a  circle  is  1,  the  circumference  will  bt 
3.1416.  Hence,  if  we  know  the  diameter,  we  may  find  the  circum- 
ference by  multiplying  by3A4\6  ;  or,  if  we  know  the  circvmfrrnice^ 
we  may  find  the  diavuUr  by  diiidin^^  by  3. 1416. 


MENSURATION.  ^05 


EXAMPLES. 


1.  The  diameter  ot  a  circle  is  4,  what  is  the  circumference? 

2.  The  diameter  of  a  circle  is  93,  what  is  the  circumference? 

3.  The  diameter  of  a  circle  is  20,  what  is  the  circumference? 

4.  What  is  the  diameter  of  a  circle  whose  circumference  is  78.54  ? 

5.  What  is    the  diameter  of  a   circle  whose  circumference  is 
n652.1944? 

6.  What  is  the  diameter  of  a  circle  whose  circumference  is  6850  ? 

324.  To  find  the  area  or  contents  of  a  circle, 

Multiply  the  square  of  the  diameter  by  the  decimal  .7854   iBk.  V. 
Prop.  XII.  Cor.  2). 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  6  ? 

2.  What  is  the  area  of  a  circle  whose  diameter  is  10? 

3.  What  is  the  area  of  a  circle  whose  diameter  is  7  ? 

4.  How  many  square  yards  in  a  circle  whose  diameter  is  3j  feet  ? 

325.  A  sphere  is  a  figure  terminated  ^rffj^jMi 
by  a  curved  surface,  all  the  parts  of  which 
are  equally  distant  from  a  certain  point 
within  called  the  centre.  The  line  AB 
passing  through  its  centre  C  is  called  the 
diameter  of  the  sphere,  and  AC  its  radius. 

326.  To  find  the  surface  of  a  sphere, 

Multiply'the  square  of  the  diameter  by 
3  1416  [Bk.  VIII.  Prop.  X.  Cor). 

EXAMPLES. 

1 .  What  is  the  surface  of  a  sphere  whose  diameter  is  12  ? 

2.  What  is  the  surface  of  a  sphere  whose  diameter  is  7  ? 

3.  Required  the  number  of  square  inches  in  the  surface  of  a 
sphere  whose  diameter  is  2  feet  or  24  inches. 

327.  To  find  the  contents  of  a  sphere, 

Multiply  the  surface  by  the  diameter  and  divide  the  product  by  % 
the  quotient  will  be  the  contents.  {Bk.  VIII.  Frop.  XIV.  Sch.  3). 

EXAMPLES. 

1 .  What  are  the  contents  of  a  sphere  whose  diameter  is  1 2  ? 

2.  What  are  the  contents  of  a  sphere  whose  diameter  is  4  ? 

3.  What  are  the  contents  of  a  sphere  whose  diameter  is  14in.  ? 

4.  What  are  the  contents  of  a  .sphere  whose  diauieler  is  ijft. 


306 


MENSUltATlON. 


328.  A  prism  is  a  figure  whose  ends  are  equal 
plane  figures  and  whose  faces  are  parallelograms. 

The  sum  of  the  sides  which  bound  the  base  is 
called  the  perimeter  of  the  base,  and  the  sum  of  the 
parallelograms  which  bound  the  solid  is  called  the 
convex  surface.  \^  i 

329.  To  find  the  convex  surface  of  a  right  prism, 

Multiply  the  perimeter  of  the  base  by  the  perpendicular  height,  and 
the  product  will  be  the  convex  surface  {Bk.  Vll.  Pro]).  1). 

EXAMPLES. 

1.  What  is  the  convex  surface  of  a  prism  whose  base  is  bounded 
by  five  equal  sides,  each  of  which  is  35  feet,  the  altitude  being  26 
feet? 

2  What  is  the  convex  surface  when  there  are  eight  equal  sides, 
each  15  feet  in  length,  and  the  altitude  is  12  feet? 

330.  To  find  the  solid  contents  of  a  prism. 

Multiply  the  area  of  the  base  by  the  altitude.,  and  the  product  will 
be  the  contents  (Bk.  VII.  Prop.  XIV). 

EXAMPLES. 

1.  What  are  the  contents  of  a  square  prism,  each  side  of  the 
square  which  forms  the  base  being  15,  and  the  altitude  of  the 
prism  20  feet? 

2.  What  are  the  contents  of  a  cube  each  side  of  vfhich  is  24 
inches  ? 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which  the 
length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and  height  or 
thickness  2  feet  6  inches  ? 

4.  How  many  gallons  of  water  will  a  cistern  contain  whose  di- 
mensions are  the  same  as  in  the  last  example  ? 

5.  Required  the  contents  of  a  triangular  prism  whose  height  ifc 
lO  feet,  and  area  of  the  base  350  ? 


331  •  A  cylinder  is  a  figure  with  circular 
ende.  The  line  EF  is  called  the  axis  or  alti- 
tude, and  the  circular  surface  the  convex  sur- 
face of  the  cylinder. 


MENSURATION. 


ao7 


332.  To  find  the  convex  surface, 

Multiply  the  circumference  of  the  base  by  the  altitude^  and  the  pro- 
duct will  be  the  convex  surface.     [Bk.  VIII.  Prop.  I). 

EXAMPLES. 

1.  What  its  the  convex  surface  of  a  cylinder,  the  diameter  of 
whose  ba.se  is  20  and  the  altitude  50  ? 

2  What  is  the  convex  surface  of  a  cylinder,  whose  altitude  is 
14  te^t  jind  the  circuniference  of  its  base  8  feet  4  inches  ? 

3.  What  is  the  convex  surface  of  a  cylinder,  the  diameter  of 
whose  base  is  30  inches  and  altitude  5  feet  ? 

333.  To  find  the  contents  of  a  cylinder, 

Multiply  the  area  of  the  base  by  the  altitude :  the  product  will  be 
the  contents.     (Bk.  VIII.  Prop.  II). 

EXAMPLES. 

1 .  Required  the  contents  of  a  cylinder  of  which  the  altitude  is 
12  feet  and  the  diameter  of  the  base  15  feet? 

2.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  20  and  the  altitude  29  ? 

3.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  1 2  and  the  altitude  30  ? 

4.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whoso 
base  is  1 6  and  altitude  9  ? 

5.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  50  and  altitude  15  ? 


334.  A  pyramid  is  a  figure  formed  by 
several  triangular  planes  united  at  the 
same  point  S,  and  terminating  in  the 
different  sides  of  a  plain  figure  as 
ABODE.  The  altitude  of  the  pyramid 
is  the  line  SO.  drawn  \)erpendicular  to 
the  base 


335.   To  find  the  contents  of  a  p>Tamid, 

Multiply  the  area  of  the  base  by  the  'iltitude.  and  divide  the  pf 
dud  by  3       {Bk    VII    Proy    XVI f) 


308 


MENSURATION. 


EXAMPLES. 

1.  Required  the  contents  of  a  pyramid,  of  which  the  aren  of  the 
base  is  95  and  the  altitude  15. 

2.  What  are  the  contents  of  a  pyramid,  the  area  of  who&e  ba.se 
is  260  and  the  altitude  24  ? 

3.  What  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  207  and  altitude  18? 

4.  What  are  the  contents  of  a  pyramid,  the  area  of  whose  bas* 
is  403  and  altitude  30  ? 

5.  W^hat  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  270  and  altitude  16?  ' 

6.  A  pyramid  has  a  rectangular  base,  the  sides  of  which  are  25 
and  12:  the  altitude  of  the  pyramid  is  36:  what  are  its  con- 
tents ? 

7.  A  pyramid  with  a  square  base,  of  which  each  side  is  30,  has 
an  altitude  of  20  :  what  are  its  contents  ? 


336.  A  cone  is  a  figure  with  a  circular 
base,  and  tapering  to  a  point  called  the 
vertex.  The  point  C  is  the  vertex,  and  the 
line  CD  is  called  the  axis  or  altitude. 


337.   To  find  the  contents  of  a  cone, 

Multiply  the  area  of  the  base  by  the  altitude^  and  divide  the  pro- 
duct by  3.     {Bk.  VI 11.  Prop.  V). 


EXAMPLES. 

1,  Required  the  contents  of  a  cone,  the  diameter  of  whose  base 
is  5  and  the  altitude  10. 

2  What  are  the  contents  of  a  cone,  the  diameter  of  whose  base 
is  18  and  the  altitude  27  ? 

3.  What  are  the  contents  of  a  cone,  the  diameter  of  whose  base 
is  20  and  the  altitude  30  ? 

4.  What  are  the  contents  of  a  cone,  whose  altitude  is  27  feet, 
and  the  diameter  of  the  base  10  feet? 

5.  What  are  the  contents  of  a  cone  whose  altitude  is  12  feet, 
and  tlie  <liarneter  <if  its  })asc  15  loot? 


GUAGING.  'SOi) 

GUAGING. 

338.  The  mean  diameter  of  a  cask  is  found  by  adding  to  the 
head  diameter,  two  thirds  of  the  difference  between  the  bung  and 
head  diameters,  or  if  the  staves  are  not  much  curved,  by  adding 
six-tenths.  This  reduces  the  cask  to  a  cylinder.  Tlien,  to  find 
the  solidity,  we  multiply  the  square  of  the  mean  diameter  by  the 
decfmal  .7854  and  the  product  by  the  length.  This  will  give 
the  solid  contents  in  cubic  inches.  Then,  if  we  divide  by  231, 
we  have  the  contents  in  gallons.     (Art.  114). 

Multiply  the  length   by  the  square   of  the  operation. 

mean  diameter,  then  by    the  decimal  -7854,     I  x  d"^  X  ' '^y3T  ~ 
and  divide  by  231.  Ixd'^x0034. 

If,  then,  we  divide  the  decimal  .7854  by  231,  the  quotient  car- 
ried to  four  places  of  decimals  is  .0034,  and  this  decimal  multi- 
plied by  the  square  of  the  mean  diameter  and  by  the  length  of  the 
cask,  will  give  the  contents  in  gallons. 

339.  Hence,  for  guaging  or  measuring  casks,  we  have  the  fol- 
lowing 

Rule. — Multiply  the  letigth  by  the  square  of  the  mean  diameter  ; 
then  multiply  by  34  and  point  off  four  decimal  places^  and  the  pro- 
duct will  then  express  gallons  and  the  decimals  of  a  gallon. 

1.  How  many  gallons  in  a  cask  whose  bung  diameter  is  36 
inches,  head  diameter  30  inches,  and  length  50  inches  ? 

We  first  find  the  difference  of  the  diameters,  operation. 

of  which  we  take  two-thirds  and  add  to  the  36  —  30=::    6 

liead  diameter.  We  then  multiply  the  square  f  of  6    =:    4 

of  the  mean  diameter,  the    length  and  34  30  +  4  =  34 

together,  and  point  off  four  decimal  places  34"^=:  1156 
in  the  product.                                                       1156x50x34== 

2.  What  is   the  number  of  gallons   in  a  196..52^aZ. 
cask  whose  bung  diameter  is  38  inches,  head 

diameter  32  inches,  and  length  42  mches  ? 

3.  How  many  gallons  in  a  cask  whose  length  is  36  inches,  bung 
diameter  35  inches,  and  head  diameter  30  inches  ? 

4.  How  many  gallons  in  a  cask  whose  length  is  40  inches,  head 
diameter  34  inches,  and  bung  diameter  38  inches? 

5.  A  water  tub  holds  147  gallons-  the  pipe  usually  brings  in 
14  gallons  in  9  minutes  :  the  tap  discharges  at  a  medium,  40  gal- 
Jons  in  31  minutes.  Now,  supposing  these  to  be  left  open,  and 
the  water  to  be  turned  on  at  2  o'clock  in  the  morning  :  a  servant 
at  5  shuts  the  tap,  and  is  solicitous  to  know  at  what  time  the  tub 
will  be  filled  in  ease  the  water  «HMitimie?!  to  ll(t\v. 


31(3 


APPENDIX. 


FORMS  RELATING  TO  BUSINESS  IN  GENERAL. 


FORMS    OF    ORDERS. 

Messrs.  M.  James  &  Co. 

Please  pay  John  Thompson,  or  order,  five  hundred 
dollars,  and  place  the  same  to  my  account,  for  value  received. 

Peter  Worthy. 
Wilmington,  N.  C,  June  1,  1855. 

Mr.  Joseph  Rich, 

Please  pay,  for  value  received,  the  bearer,  sixty-one 
dollars  and  twenty  cents,  in  goods  from  your  store,  and  charge  the 
same  to  the  account  of  your 

Obedient  Servant, 

John  Parsons. 
Savannah,  Ga.,  July  1,  1855. 


FORMS  OF    RECEIPTS. 

Receipt  for  Money  on  Account. 

Received,  Natchez,  June  2d,  1855,  of  John  Ward,  sixty  dollars 
on  account. 


$60,00  John  P.  Fay. 

Receipt  for  Money  on  a  Note. 

Pi.eceived,  Nashville,  June  5,  1856,  of  Leonard  Walsh,  six  hun- 
dred and  forty  dollars,  on  his  note  for  one  thousand  dollars,  dated 
New  York,  January  1,  1855. 

S640.00  J.  N.  Wefks. 

NOTES. 

1.  A  Note,  or  as  it  is  generally  called,  a  promissory  note,  is  a 
positive  engagement,  in  writing,  to  pay  a  given  sum  at  a  time 
specified,  either  to  a  person  named  in  the  note,  or  to  his  order,  oi 
to  the  bearer. 

2.  By  mercantile  usage  a  note  does  not  really  fall  due  until  the 
expiration  of  3  days  after  the  time  mentioned  on  its  face.  The 
Ihrctj  iuldiiioiiiil  days  aie  called  days  of  ^tnce. 


AITENDIX.  311 

When  the  last  day  of  grace  happens  to  be  Sunday,  or  a  holiday, 
such  as  New  Years,  or  the  Fourth  of  July,  the  note  must  be  paid 
the  day  before :  that  is,  on  the  second  day  of  grace. 

3.  Theie  are  two  kinds  of  notesMiseounted  at  banks  :  1st.  Notes 
given  by  one  individual  to  another  for  property  actually  sold — 
these  are  called  business  notes,  or  business  paper.  2d.  Notes  made 
for  the  purpose  of  borrowing  money,  which  are  called  accommo- 
dation notes,  or  accommodation  paper.  Notes  of  the  first  class  are 
much' preferred  by  the  banks,  as  more  likely  to  be  paid  when  they 
fall  due,  or  in  mercantile  phrase,  "when  they  come  to  maturity." 

FORMS    OF    NOTES. 

No.  1.  Negotiable  Note. 

$25^  Providence,  May  1,  1856. 

For  value  received  I  promise  to  pay  on  demand,  to  Abel 
Bond,  or  order,  twenty-five  dollars  and  50  cents. 

Reuben  Holmes. 


Note  Payable  to  Bearer. 
No.  2. 
$875,39.  St.  Louis,  May  1,  1855. 

For  value  received  1  promise  to  pay,  six  months  after 
date,  to  John  Johns,  or  bearer,  eight  hundred  and  seventy-five 
dollars  and  thirty-nine  cents. 

Pierce  Penny. 


Note  by  two  Persons. 
No.  3. 


$659,27.  Buffalo,  June  2,  1856. 

For  value  received  we,  jointly  and  severally,  promise  to 
pay  to  Richard  Ricks,  or  order,  on  demand,  six  hundred  and  fifty- 
nine  dollars  and  twenty-seven  cents. 

Enos  Allan. 

John  Allan. 


Note  Payable  at  a  Bank. 
No.  4. 


$20,25.  Chicago,  May  7,  1856. 

Sixty  days  after  date,  I  promise  to  pay  John  Anderson, 
or  order,  at  the  Bank  of  Commerce  in  the  city  of  New  York, 
twenty  dollars  and  twenty-five  cents,  for  value  received. 

Jesse  Sroitts. 


312  APPENDLX. 


REMARKS    RELATING    TO    NOTES. 

1.  The  person  who  signs  a  noie,  is  called  the  drawer  or  vuiker 
of  the  note ;  thus,  Reuben  Holmes  is  the  drawer  of  Note  No,  1. 

2.  The  person  who  has  the* rightful  possession  of  a  note,  is 
called  the  holder  of  the  note. 

3.  A  note  is  said  to  be  negOiiable  when  it  is  made  payable  to 
A  B,  or  order,  who  is  called  the  payee  (see  No.  1).  Now,  if  Abel 
B'^nd.  to  whom  this  note  is  made  payable,  writes  his  name  on  the 
back  of  it,  he  is  said  to  endorse  thu  note,  and  he  is  called  the  en- 
doiser ;  and  when  the  note  becomes  due,  the  holder  mu.<:t  first 
demand  payment  of  the  maker,  Reuben  Holmes,  and  if  he  declines 
pacing  it,  the  holder  may  then  require  payment  of  Abel  Bond,  the 
endorser. 

4.  If  the  note  is  made  payable  to  A  B,  or  bearer,  then  the 
drawer  alone  is  responsible,  and  he  must  pay  to  any  person  who 
hold:  the  note. 

5.  The  time  at  which  a  note  is  to  be  paid  should  always  be 
named,  but  if  no  time  is  specified,  the  drawer  must  pay  when  re- 
quired to  do  so,  and  the  note  will  draw  interest  after  the  payment 
is  demanded. 

6.  When  a  note,  payable  at  a  future  day,  becomes  due,  and  is 
not  paid,  it  will  draw  interest,  though  no  mention  is  made  of  inter- 
est. 

7.  In  each  of  the  States  there  is  a  rate  of  interest  estab-i.'^l.ed  by 
law,  which  is  called  the  legal  interest,  and  when  no  rale  is  speci- 
fied, the  note  will  always  draw  legal  interest.  If  a  rale  higher 
than  legal  interest  be  taken,  the  drawer,  in  most  of  the  States,  is 
not  bound  to  pay  the  note. 

8.  in  the  State  of  New  York,  although  the  legal  irterest  is  7 
per  cent,  yet  the  banks  are  not  allowed  to  charge  over  6  per  cent, 
unless  the  notes  have  over  63  days  to  run. 

9.  If  two  persons  jointly  and  severally  give  their  note,  (see  No. 
3,)  it  may  be  collected  of  either  of  them. 

10.  The  words  "  For  value  received"  should  be  expressed  in 
every  note. 

1 1 .  When  a  note  is  given,  payable  on  a  fixed  day,  and  in  a  spe- 
cific article,  as  in  wheat  or  rye,  payment  must  be  offered  at  the 
specified  time,  and  if  it  is  not,  the  holder  can  demand  the  value  in 
money. 

A    BOND    FOR    ONE    PERSON,    WITH    A    CONDITION. 

KNOW  ALL  MEN  BY  THESE  PRESENTS,  That  I,  James 
Wilson  of  the  City  of  Hartford  and  State  of  Connecticut^  am  held 
and  firmly  bound  unto  John  Pickens  of  the  Town  of  Waterhury. 
Cwnty  of  New  Haven  atul  State  of  Conneclicaty  in  the  sum   uj 


AJFFENDIX. 


ai3 


Eighty  dollars  la^A-ful  money  of  the  United  States  of  America,  to 
be  paid  to  the  said  John  Pickens^  his  executors,  administrators,  or 
assigns :  for  which  payment  well  and  truly  to  be  made  I  bind 
myself^  my  heirs,  executors,  and  administrators,  firmly  by  these 
presents.  Sealed  with  my  Seal.  Dated  the  Ninth  day  of  March^ 
one  thousand  eight  hundred  and  thirty-eight. 

THE  CONDITION  of  the  above  obligation  is  such,  that  if  the 
above  bounden  James  Wilson^  his  heirs,  executors,  or  administra- 
tors, shall  well  and  truly  pay  or  cause  to  be  paid,  unto  the  above- 
named  John  PickenSj  his  executors,  administrators,  or  assigns,  the 
just  and  full  sum  of 

[Here  insert  the  condition.] 
then  the  above  obligation  to  be  void,  otherwise  to  remain  in  full 
force  and  virtue. 

Sealed  and  delivered  in 
the  presence  of 

John  Frost.  } 

Joseph  Wiggins^  J 


James  Wilson. 


Note.  The  part  in  Italic  to  be  filled  up  according  to  circuiri- 
ftance. 

If  there  is  no  condition  to  the  bond,  then  all  to  be  omitted  after 
and  including  the  words,  "  THE  CONDITION,  &c." 

BOOK-KEEPING. 

Persons  transacting  business  find  it  necessary  to  write  down 
the  articles  bought  or  sold,  together  with  their  prices  and  the 
names  of  the  persons  to  whom  sold. 

Book-keeping  is  the  method  of  recording  such  transactions  in  a 
regular  manner. 

COMMON  ACCOUNT  BOOK. 

The  following  is  a  very  convenient  form  for  book-keeping,  and 
requires  but  a  single  book.  It  is  probably  the  best  form  of  a  com- 
mon Account  Book. 


J.  BELL. 


Dr. 


J.  BELL. 


Cr. 


1846. 
lune  1 

"    6 
July  9 


To  5  cords  of  wood, 
at  $1,75  per  cord, 
To  1  day's  work, 
To  Uu.  ol  rye,  at  62 
cents  per  bu. 


$ 

c. 

1846. 

8 

75 

July  6 
"  10 

1 

00 

"  20 

2 

48 

Aug.  1 

12 

23 

By  shoeing  horse, 
"  mending  sieigh, 
"  ironing  wagon, 
"  Cash  to  balance. 


^   c. 

1  00 
3  20 

5  12 

2  86 


12  2.' 


U 


314 
ANSWEES. 


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317 


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86. 
86. 
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9 
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11 
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88. 
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3 

4 
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12 
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1 
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3 

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4 
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8 
9 

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15 
16 
17 
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19 
20 
21 
22 
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15 

28 

324 

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2  '  30183/a/-. 

3  I  84226>y. 


4  I  39l679yar. 

5  £84 


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2  I   \2yd. 
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53796602/^5. 
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21^  mi. 


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8 

1   Ana.  16/ja.  32//«.  96wa.  \28yuu 

2Wqr.  32qr.   2Sqr. 

3  \3qr.  Aqr.  5qr.  9qr.  \Oqr. 


102. 
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3\980?ia. 

4  623?^.a. 

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6 

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103. 
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1 

(  288in.  \32in. 
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2  40P.    120P.   640ii.    12807t. 

3  160P.  320P.   9,jd. 

104. 

1 

104. 

2 

104. 

3 

104. 

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4  762300. 

5  260sq.  ft.  16.9^.  in. 

6  93.4.  2R.  12P. 

7  35ilf.  563A  IR.  19P. 

8$12584,25. 
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105, 
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105. 
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105. 
105. 

106. 

106. 
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j  1728  0/^.  m.  3i56Cu.in. 

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24  C.fl.  40  C.ff:  48  C.  ft. 
256  S. ft.  64  S.  ft.  32 S.  ft. 


5|2C.  ijd.  3C.  yd. 
6I3T. 
7|2T.  4T. 
8\8C.  120.  16C. 
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592704. 

200  c.  ft.  3200  S.  ft. 

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21870  cord.s-4C.ft. 
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107. 
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136,64. 


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212bu. 

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110. 

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4      I2^r. 

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112. 

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112. 

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112. 

3  148340^r.                      10  J 

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113. 

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114. 

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7    .739 1 8 -r. 

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114. 

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114. 

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115. 
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115. 
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1 
2 
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3 
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116. 
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P. 

EX. 

116. 

2 

116. 

3 

116. 

4 

116. 

1 

116. 

2 

ANS. 


360 w.  240m.  300m. 
120°  180°  210°  240° 
4°  12°  3s.  5s.  6s. 
10765' 
2592000" 


117. 
117. 
117. 
117 
117 


57953Ar. 

10800' 

1296000  Cw.  in, 

£714 

3  T.  Icwt.  20lb. 


6 

7 

8 

9 

10 


EX. 

3" 
4 
5 
6 

7 


T 

Ic.  5s. 

39468005ec. 

921625.sec. 

2°  23'  9" 


9fe  8  5    13   2  9   I9^r. 

lib.  Qoz.  ^pwt.  \9gr. 

340\61gr. 

S\g.  Is. 

201 E.  E.  3qr. 


118. 

11 

118. 

12 

118. 
118. 

13 

118. 

14 

118. 

15 

118. 

16 

118. 

17 

118. 

18 

118. 

19 

118. 

20 

118. 

21 

118. 

22 

118. 

23 

3320  half  pints. 
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I  6 A.  IR.  24P. 

1  pound. 

67953hr. 

7s.  15°  24'  40" 

12  cords. 

1244 160  Cm.  in, 

\20962)k. 

311yd.  2qr. 

48916gi. 

41 8002432sq,  in. 

15359fa?: 


4rd, 


84mi.    3fur. 

3yd.  2ft. 

5A  3R.  35P. 

2ft.  5in. 
197111025ikr. 
26880  times. 
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827 

4  fniles. 
40  yards. 

57no.3wk.  5da.  16Ar 
1008  bottles 
110592 
38  casks. 


3iyd, 


119. 
119. 


36 
37 


17097yy^  times. 
10132992005ec. 


38 
39 


248  miles 
$39,879 


120. 
120. 
120. 
120. 
120. 
120. 

2 
3 
4 
5 
6 
7 

£1377  45.  l\d. 
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8 
9 

10 

20  3    1  9   10^/. 
j  19cwt.  2qr.   18lb, 
j  l5oz.  lldr, 
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\  20lb.  2oz. 

121. 
121. 
121. 
121. 
121. 
121. 
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11 
12 

13 

14 
15 

\6cwt.  2dr. 

j  432i.     2mi.    4fur. 

\  39rd.  4yd. 

j  \fur.    34rd.    l^yd. 

\  \ft.  4in. 
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16 

17 

18 
19 

j  184E.E.  4qr.  2?ia, 

263sq.yd.     5sq.  ft 

1 1 657.  in. 

21 M.  211  A.  \R. 
■  OP.  24I.S.  yd. 
159 A.  2R.  5P. 

Ml 


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ANSWERS. 


121. 

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122. 
122. 
122. 
122. 
122. 
122. 
122. 
122. 
122. 

123^ 
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123. 
123, 
123. 
123. 
123. 
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20 
23 
24 

25 

126 
27 


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176C.  yd.  \SC.ft. 
614 C.  iM. 


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22 


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90  C.  ]06C.ft. 
151  C.  3C.//. 


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2596-;i.  \2bu.  Opk.  Oqt. 

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Akr. 


28 
29 
30 

1 


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59" 

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3" 

291/Z*.  6o2.  15/^. 

22-r. 


244/A.  5o2.  4;^^.  'dgr. 

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l()lb.  loz.   7 dr. 

41  T.  OcwL  Sqr.  17 lb. 

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336.4.     111.     31  P. 

2 [OSq.fi.  1 36 Sq.  in, 
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1  (JSbu.  Opk.  2qt. 


9 
10 
11 
12 
13 
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15Sbu.  Opk.  Aqt. 
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3lb.  loz.  1  Ipivi.  17 gt 
322mi.6fur.  Urd. 
lOOA.  IR.  13P. 


124, 

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125 
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126. 
126. 
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126. 
126, 
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2 
3 

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4 

17 yr.  Imo.  3da. 

6 

7 

1 

2 
3 
4 
5 

12yr.  3mo.  26da.  22hr. 
30yr.  Imo.  29da.  12/ir. 

J  27mo.  3ivk.  Oda. 

\  20kr.  20m. 
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j  7T.  IScwt    Iqr.   Alb. 

\  Onz.  2dr. 

7 

8 

9 

10 

11 

12 

13 
11 

2m.  Afur.  21  rd. 
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1  1  B   lAgr. 
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130. 

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130. 

25 

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31 

3174  miles. 

130. 

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130. 

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33 

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132. 

11 

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23 

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13 

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24 

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132. 

14 

2°  34'  16" 

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15 

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29    1    Algal  3qt.  Xf^pt. 

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15' 

3 

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■J2i 


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35 

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133. 

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133. 

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135. 

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7 

4/ 

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135. 

2 

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8 

4/ 

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135. 

3 

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9 

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135. 

4 

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j$175 

1  4- $125 
$10500 
160yd. 

$18 

$83,331 
7ida. 
3  men 
5^mo. 

30 
31 

36  men 
(A  21 
,  B  15 
^  C24 

hogs 

212. 
212. 
212. 
212. 

32 

33 
34 

j  ^'s$126.  ^'5 
1  $117.  C's  $72. 

5ibar. 

U70  gal. 

35|$0,803  + 

36\10iyd. 

37|$1344 

38IJS55040 

^c,\lst.  $39,20  2d. 
'^^|$19,60  3rZ.117,60 
40  $533,331 
413  pieces 

216. 

217. 
217. 


1     36 


2  I  60  113  I  12  114  I  21  !|  5  I  24     —     — 


4-— 1 

Tfi  —  4- 


A\   2  0—1    I 

^  200_2    ! 
^'300— 1^- 


J-iA  =  4  I 
T2— T-    ' 


'fil   5    _  1 
I^^O  — 4' 

qU_— 1 


8  —  ^8- 


'^=tV- 


T32- 


210. 
219. 
219. 

220, 
220. 
220. 
220. 
220. 
221. 
221, 
221. 
221, 


308mi. 

$165 

$1381,25 


3300  jjounds. 

$61,425 

10955mi. 


1243.75 

$201,75 


1061yd, 
36000  rations. 


8hr.  20m. 
$4,861-f- 
£227  125.  \d.-\- 
$115,50 
$29,25 


171811/^. 


$40,32 

^'5  $1787,50 
J5'5  $1285,75 

20  $22,50 


19 


$1,871 
$0,154  + 
$6206,93^ 


2\\20\^bar. 

28 

$252 

j32 

21|/6. 

36 

2-3|$6 1,425 

29 

2'lyd. 

33 

$12,13 

37 

2(),55.  M, 

30 

72  Jmts 

34 

28hr. 

38 

21315bu. 

31 

37  bar. 

35 

2^  acres 

39 

$18,27 
$168,742-1- 
$108,25 
$253,125 


223. 
224. 
224. 
224. 
226. 
227. 
227. 
227. 
227. 


1  j  8  days. 


27da.   j 

72da. 

I60da.\ 


6\20hr. 
6  18fia. 
7\l6bof:s 


8,  6men 
91  134^a. 
10  276^a. 


11 

16lH. 

14 

12 

25bu. 

— 

13 

lOda. 

— 

l-^lb. 


5TJ 


jl  $45  II  2  I  150/6.  II  3  I  $99  ||  4  |  2326/^.  ||  5 


5lliwi. 


13flX 

10 

4^\da 

14 

18yr. 

11 

1112bu. 

15 

27  weav's 

12 

2f  tons 

16 

72  men 

13 

3431/^ 

17 

15Z6. 
38-|rea's 
1^2bar. 
200  more 


18 


36yd* 
long 


229. 
229. 
229, 


$857,142f  As.  $142,8571  B' s- 
$480  As.  $750  ^'5.  $675  C's. 
$1500  Mr.  W's.  $2100  Mr  Ts. 
$400^\v.  $800^'s.  $1200  G's. 


$1866,66|  As 
41    {  $1066,66f  j5'5. 
'    (  $1066,e6J  C"s. 


332 


AKSM^EieS. 


p. 

m 

230. 
230. 
230. 
230. 
230. 
230. 
230. 
230. 


EX. 


ANSWERS. 


$77  As.   $260  B's. 

$54.  A's.   $38.50  ^'5. 

j  $60,777  4- -A's.  $127,633  +  i?'5.  $233,387 -fC'5. 
I  $328,201 +i)'5. 

$lC66.66f  A's.   $3888.88f  B's.   |9444.44|  C's. 

^'5  =$273,365  nearly.    ^'5=$476,635. 

Fuller's  $1808,8669+,  jBro«^;?z'5  $1596,0591  + 
Dctters  $1995,0738+,  The  remainders  added 
will  give  the  exact  'proof. 


232. 
232. 
232. 
232. 
232. 
232^ 

233. 
233, 
233. 
233. 
234. 


I|$16,25 
2  19,50i/6^. 
3|39,375cw;t 

$2,375 

155,48wz. 

5  oxen. 


\^\ 


$8,93 

18,06  sheep 
$18,5487 
280  cows 
892,5  tons 
1015/6. 


13l$2109,0392 


$75 

$229,08 

$350 

$375 

$694,232 


$42,60 
4326«r. 
A.2hhd. 

$24.25 


23 
24 
25 


$10,80 

\  26^per  ct.  left 
\  —  $3333,33^. 

$1304,75 


1 

,25 

5 

2 

,50 

6 

3 

,40 

7 

4 

,20 

8 

T3' 


,38 
,05 

,03 


I'V^. 


10 


.66  II   11  I  .20  II    12  I  .20   II    13  |  .121 


235. 
236. 

238. 
238. 
239. 
239. 
239. 


2  I  $24862,50  ||  3  |  $233,75  ||  4  |  $8443,75  ||  5  j  $14700 


200  shares. 


il,06J. 
;0,75  loss. 


237.  I  10  I    80  shares. 


$0,966  + 
$1,00 


$112,50 

$208,4375 


7  ,  $2,054 
&  I  26p€rct. 


9 
10 


1 8  per  ct. 
j  $13  whole  g'n 
\  —2^  perct. 


11 
12 
13 


$1,025 

$1,034 

$2,2l6f 


14 
15 


19^^/. 


,21^^. 


240. 
240. 
240. 
240. 


1|$43,77 
2|$1312,50 

$237,60 
^'  ^  $158,40 


$210 

$607,50 

$1381,80 


7i$504 


$450 

$1320 

$142,95 


11|$1800— $45 
12|  $47,624  + 
13$9558,437+ 
14|$6500 


242. 
242. 
242. 
242. 
243. 


$39 
$266 
$4446,75 
$642,60 


$427,50 
$9,5067 
$331,1511 

$1158,0668 


$183,9705 
$4454,857 
$30455,0224 
$95,229+ 


$121,325 
1315,389 

221,075 
1290,798 


I  2  I  $10,8012  II  3  I  $2,728+       —         —         ^ 


AMSWERS. 


383 


p. 

EX. 

ANS. 

EX. 

4 

ANS.               I 

:x. 

ANS. 

244. 

2 

$309,5634 

$30,5598 

6 

$64,5792 

244. 

3 

$35,1485-1- 

5 

$14,0979 

- 

245. 

7 

$76,2433 

13 

$190,148 

19 

$600,445 

245. 

8 

$194,6177 

14 

$3286,40 

20 

$44,2893 

245, 

9 

$328,32 

15 

$6322,8825 

21 

$107,001 

245. 

10 

$1004,6976 

16 

$7500,60 

22 

$3120,203 

245. 

11 

$1183,6935 

17 

$75,04 

23 

$9051,668 

245. 

12 

$1445,2388 

18 

$218,88 

24 

$4968,9975 

246. 

25 

$141,8136 

31 

$94,265 

37 

$217,5116 

246. 

26 

$272,80 

32 

$245,4896 

38 

$6214,14 

246. 

27 

$39,9274 

33 

$76,966 

39 

$856,686 

246. 

28 

$928,0686 

34 

$33.3232 

40 

$383,3808 

246.! 

29 

$529,925 

35 

$28761,776 

41 

$188,0292 

246.1 

30 

$31,2681 

3i 

$5678,068 

42 

$2418,465 

P17 

1    2 

£15  2?   8ld 

A 

£'>6>  10?    11/7 

217 

u 

£24  185.  S^d-^ 

5 

£331  Is.  6d. 

249.1!   2  1  $860,4194  ||  3  | 

$167,983-f- 

250.1 

l|$i 

)50||2|7;?er 

ct.\\: 

3!5?/r.i|4|$225||5|7 

,331— 7?/;-.  ^nw. 

251. 
251. 

2 

3 

$19,101 

$36,50  + 

4 
5 

$404,0625 
$291,60 

6 

7 

$211,456 

$165,775 

252.  i 

8|$171,597l||9i$118,528||10|$315,2438|i  11  |  $15-2,408 

253 
254. 
254. 
254. 
254. 
254. 
255. 
255. 
256. 
257. 
258. 

m 

260. 
260. 


$1750  present  value.  ||  2  |  $1565,402-|-  pres.  val. 
8   %ob'6'd ,A01 -[- pres.  val. 


$9677,50   +  pres.  val 
£223  6s.Qd.  discount. 


5    $5620,176+ j^-z-ei.mZ. 


$702,485 

$1,94     difference. 


$2109,236  + 
$2763,694  + 
$4000 
$6,473+  loss. 


il|$6,329l 
2$10,50 


$15240,54 

$5,8408 


$3393,504 
$29.0096 


7  $122,81+ 


8  I  $341.709+11  1  I  $344,66+  i|  2  |  $5734,32  + 
3|  $695,64  II  4  |  $118,85+  ||  5  |  $1740,60  ||6  |  $376,46  + 

2  ll2mQ.!|3|8mQ.  22^da.\\^^mo.\\6\\T/\\da.\\^^mo. 
\7\61j\da.  9lh  day  of  March.  -—  


9 
10 


60j-9^y/a.  or  Aug.  ^Ist. 
Qfuo.  (jda. 


49jWda.  or 
Jan.  25th 


262.|i  3  I  $12,25  ||  4  |  $6,25 


334 


ANSWERS. 


265. 

265. 
265.! 

265.! 
265.1 


$426,416 
£1073  185.  lid. 
$2033.4894- 
£389  6s.  2\d. 
^2551,733 


ANS.' 


(  £21  5s. -£25  14.S-.  3r/.  £30 
175.  l6/.-(--£41  25.  ^d.-^ 
(  £38  ll5.  4irZ.-£23  V^s.  U\d 
i  $250-1250425042004250 
\  $516,66^4250. 


266. 

266. 

1 

2 

$3720,937 
$8668,935 

3 
4 

$6748,60 
$4583,94  + 

5      $3643,875 

267.11   2  1  $5944,791   ||  3  |  $9226,061+            

268.|i   2  1  $1270,428    ||    3  |  $2016,11    |j   4  |  $16975,775 

270.111 1-^28 12,50||2|$423,36    j|  3  |  $251,45+  ||  4  |  $1457,75 

271.11 1  1  35.||2|  846-^5. +  ||3|  288  +  6-^5.||4|20|m'm/5.||5|73'' 

274. 

4 

1/6.-1/6.-  3/6. 

274. 

5 

3  0/  16.  2  0/ 18.  3  of  23.  5  of  24 

274. 

6 

Zgal.  at  105.-3  at  145.-4  at  2\s.  4  at  245. 

274. 

1 

4  «•«/.  at  55.-8  at  55.  6r/.-8  a^  65. 

274. 

2 

;46^^.  TF.  28bu.  RA^bu.B.  2Sbu.  0. 

274. 

3 

96bu.  W.  \2bu.  R.  \2bu.  B.  \2bu.  0. 

275. 

4 

40^a/.  F.  bOgal.  E.  20gal.  sjririts. 

275. 

1 

10  of  \st.  10  of  2d.  30  of  3d. 

275. 

1 

36/6.  at  Ad.  36  at  6d.  36  a^  \0d.  36  aj5  12d 

275. 

2 

2  If  of  each. 

275. 

!  3 

4  each  of  the  \st.  three  and  30  0/"  15  carats  fine. 

276. 

1 

12=1 

9    .    9*  =  6561 

276. 

2 

13           1 

10 

16^=1048576 

276. 

3 

11 

206  =  64000000 

276. 

¥  -5TT- 

12 

2252  =  50625 

276. 

4 

5    —TZ' 

13 

21672  =  4695884 

276. 

5 

92  =  81 

14 

3213  =  33076161 

276. 

6 

123=1728 

15 

215*  =  213675062.5 

276. 

7 

1253=1953125 

16 

=  610437195439771 

276. 

8 

163=4096 

17 

96  =  531441 

276. 

18 

360492=1299530401 

282. 

1  1.73205  + 

66031 

11 

.05 

16 

3.12249 

282. 

23.31662  + 

7J4698 

12 

.01809 

17 

0.71554 

282.1 

3  32.695  + 

8157  19  + 

13 

.0321 

18 

0.64599  + 

2^^2.1 

1  1506.23  + 

9'69.247  + 

14 

2.104 

19!f 

282.! 

5  2756.22  + 

iO|2.oyi  + 

15 

2.91547  + 

20:f 

ANSWERS. 


335 


p. 

EX. 

2 

2 

ANS. 

EX. 

4 

ANS. 

EX. 

6 

ANS. 

284. 

284.1 

25ft. 

1  2.6  4  9r<i. 
1  2  6.4  9rrf.+ 

85 
97.75mi.-\- 

43.81  ?-^.  4- 
82  partners 

285.  i 

7 

1   62 

trees 

11 

8 

1   \60rd 

.    II 

9   1   90// 

.  II  10 

1 

A.94ft. 

288. 
288. 

i 

73 
179 

I 

319 
439 

5  638  1 

6  364  1 

1 
2 

54       3 
955     4 

2.35 
.909 

0 
6 

.707 
1.505 

289. 
289.1 

289.1 

1 
2 

3 

5 
31 

3^- 

4; 
5 
6 

.269-f 
.8^.2  + 
.873 

1 
2 
3 

17 

28-4704 

16.197/^.+ 

4 
5 

6 

14.58//.+ 

1728 

I2ft. 

7 

b>. 

290.1 

tt  1   268.0832  1 

9  1  2ft.  iin. 

10  \2/L\\  11    1    V2ft. 

292.1 

1    1    $1,53  II    2    1    ^212  II  3  1  40  II  4    j    2  ||  5    |    9yr. 

293.1 

1  1  47jr.    II    2  1  67m.    II    3  1  Acents. 

294.||2i7 

8/zwe5|!3|34-162||4i$l,3 

5||5|175wz.||6|5mi.l300?/rZ. 

296. 

|2!£2  2.9.  8d.\ 

1  31  4  II  4  I  78732  ||  5  |  $25600  ||  6  |  $61,44 

297. 

1 

6560 

3 

381 

5     $196>83-$295,24 

297. 

2 

254 

4 

£204  155 

6     $48C0-$9450 

298. 

1 

^978 

6 

3 

Ill 

J  213,3125 
211,6875 

'  15 

$3567 

298. 

2 

516?*. 

7 

1  7  Jy(/5. 

16 

288 

298. 

3 

S80,71 

8 

it- 

12 

nandii. 

17 

$4717 

29  S. 

4 

5467/ir. 

9 

42V 

!l3 

9.04 

18 

137 

291 

0 

$26,25 

10 

120  we/i. 

'14    4i.                1 

— 

299. 

19 

10,^^7_  planks. 

29  879000 

299. 

20 

$3800 

30:792 

^99. 

21 

80yds. 

,,     (  Imi.  Ofur.  33r(L 
31    |l5iA' 

299. 

22 

\  ^'.9 128.  ^'.s  $60. 
1  C's  $32. 

299. 

32  62  years 

299. 

23 

71  days. 

33  4 

299. 

24 

37no. 

(  $2450   l5i. 

2^9. 

'''•'i 

\^\¥t 

$986,86-^     warth 

34    \  $3681   2^/. 

299. 

/  $4294,50  3d, 

299. 

26|:i 

35  £408  saves 

299. 

^«  3  l5^.$16C.2J.$120. 
"^'n  3(/.  $140  II  28  1  50.  . 

36\23^bu. 

299. 

37J2400 

300. 

38  3  o'clock 

42 

32^^^^. 

48 

j  62yr.  Umo. 
\20da,  IOU7 

BOO. 

39 

300  7nen 

43 

$34782,608 

300. 

40 

864 

44 

$3,653 

49 

Hh 

300. 

r.4'5  $2364 

45 

34  J  per  ct. 

50 

]\\yr. 

300. 

41 

B's  $1182 

46 

$7816,0914- 

51 

22500  biicks 

300. 

1  C".<f  $788 

47 

57  pieces 

52 

21.1  ft. 

300. 

TV  >.■    *¥;'i<i/l 

-       1 

53 

r    ,.,.,. ,,,f^ 

M//   0 

386 


ANSWERS. 


301. 

301. 
301. 
301. 
301. 
301. 
301. 
301. 
301. 
301. 


ANS. 


2250  men 

(i|>196.83  last  term. 
\  $29o,24  tvhole  am't 
(  946m.  wheat.  1 2  rye. 
\  12  barley.  12  oats. 

1 6l|zVi. 

356,25 

$8640 

$1,20 

lost  4  pence. 


65 


ANS. 


4  f/dZ?/5 

240  hours 

A21-B  8}-C42days. 

^'5=$l94,801f 

J5'.«=$129,87yiy. 
C"s-^$97,40ff 
D'5^$77,92if 
$1020,66 
$8925,544  + 


302. 
302. 
302. 
302. 
302. 
302. 


50ft. 

9mi.  6fur.  34:rd.-\- 
(  daughter  $780.  so?i\ 
(§3120.  m>  $1560. 

76mi.-l292mi. 

4//r.  11  wo.  22.8c?a. 


73 

74 

75 

76 
77 


4yds. 

^423,36 

($920,20  l5i.  $2760,60 
\2d.  552 L20  3d. 

3hr.  20m. 

69^mi.f^7n  N.  Haven. 


303. 
303. 


4006^.  yd. 
30A. 


2 A.  3R.  15P. 
109A  lii.  28P. 


12A.  3R. 
3A.  3.R. 


24P. 
25P. 


304.1 
304 
304. 
304. 


oA. 

\0A. 

7 A.  2R. 

6A.  OR.  12P 


8  32520s-^.  yd. 
1|45849.485. 
2i5A  lil.9.95P. 

'^.  5^7"6'^' 


;|230/^ 


176.031256-5^.  yi/. 
15A  OR.  33.2P. 
67 A  2R.  UP. 


305, 

305, 

305.1 

305. 

395. 

305j 

306.1 
306. 


12.5664 

292.1688 

62.8320 

25. 

3709 

2180.41  + 


1 

28.2744 

3 

2 

78.5400 

1 

3 

38.4846 

2 

4 

1.069  + 

3 

1 

452.3904 

4 

2 

153.93841 

- 

809.5616 
904.7804 
33.5104 
1436.7584 

113.0976cm.  y?; 


\\4.560  Sq.ft. 
2 1440  "     " 


I  4500. 
2|l3824cw./^. 


2\lcu.ft.\  5\3500cu.  ft. 


I  r.72  ]  3 


307. 
307. 
307. 
308. 
308. 
308. 

309. 
309, 


3141.6 

\l0.666Sq.ft.-j- 
6654.86 Sq.  in. 


212058 
9110.64 
3392.928 


|18G9.5616 
29452.5 


475 
2080 


4030 
1440 


6000 
65.45 


1242       6      3600       2      2290.2264     6      706.86 


3141.6 

706.86 


19G.52^aZ. 
185.0688^aZ. 


\3i\sral 
182.8Ugal. 


6hr.  3m. 


48|H 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


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demand  may  I  WHUtM 
expiration  of  1  an  period 


mm 


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DEC   5 

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m    6 '66 -7  PM 


before 


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VB   17362 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


i 


^ 


S.    BARN':S    &    CO., 

51   &  53  JOHX  JSTF    ;  I,   lnKW  YORK. 


1.    MONTEITH  AND  McNALT  Y'L 

Monteith's  First  Lessons  in  Geotrmiiliy.  (  M 
MonteitU's  Manual  of  Gc.)j:ra[)liy.  f 

2.    DA  VIES'  SEEIES   '? 

Davies'  Primary  Arithnunic.  !  !• 

Davies'  Intelloctnal  Arilhm.-tic. 
Duvies'  First  Lessons  in  Aritlimetic. 


3TIIES  '^F  GEOGRAPHY. 

■'ly's  Complete  Sclinol   Geography 
Lii  Miiiiis  ;uul  enfrraviiigs. 

ARITHMETIC  ^ 

»  os'  Ne»v  Scliool  ArithinPtic. 
i-ies'  University  Aritlimetic. 
ivies"  GniiMinar  of  Aritlimetie. 


3.    ENGLISH  GRAMMAR,  COM'^Ol^ITION,  READING,  ETC. 


Clark's  First  Lessons  in  Grammar. 

Clarlv's  Analysis  of  Enirlish  Language. 

Clark's  New'Kiiglisli  Grammar. 

WelclTs  Eiiglisii  Sontunce. 

Brookfield's  Firs;  Book  in  Composi^'"    • 

Martin's  Orrlio-pist. 

Parkers  liiiemrical  Header. 

Iligli  Scliool  Lit(M-atnre. 

Sh.'r\^oo(rs  Seif-Cnltnre  in  Elocution. 

.Par;..;r'3  Word  Builder. 

Parker  and  Watson's  series  o:'  K.  i-Ur.^ 

4.    SOIENTIFIx 
Parker's  Juvenile  Pliilosopliy,  Part  L 
Parker's  .luvenile  Philosophy,  Part  IL 
Parker's  Natural  Philosopliy,  Part  Ui. 
Porter's  First  Book  in  Science. 
Pi-  ner's  Principles  of  Chemistry. 
Hamilton's  Pliysi(dogy. 
Darby's  Southern  IJotany. 
Pa;,'e"s  Elements  of  Geology. 
Cliamhers'  Zooloicy. 
Cliamhers'  Intnidnetion  to  i': 
M(  Lntyre  on  the  G!  ibes  and  Astroijn 
6.    DAVIES'  ALGEBRA,  GEOM £. 
Davies'  Klemcntary  Alirebra. 
Davies'  Elementary  (geometry. 
Davies'  Practical  Mathematics, 
Davies"  Logic  of  .Mathematics. 
Davifs"  Legendre's  Gomevry. 
Davies'  Bourdon's  Algebra. 


I's  Dictation  Exercise 
'iniviytical  Ortiio.^rajiliy. 
■  venile  Definer.  •• 

ler's  Manual  and  Dlcti. 
,-ii  Sp^dler. 

Ill's  Paradise  Lost,   (sch.-  ed.) 

ks  Coil r.>-e  of  Time.        " 

i  ..  ■    .-■   "M-'sTask,  &c.  " 

iio\      -   i   loiiixdi's  Seasons.  " 

.. .-  \  .  ;   •  -'3  Night  Thoughts.     " 

c^i-^.tme^t. 

I  FiiltonA'xl  K.ir>! 'nan's  Book-Keeping. 


BartlettV  .^yntliotical   .Meclianic.-. 
I  Bar!  eft's  Anal.  Mechanics. 
I  li-    '  •IS  Ac'i-istics  and  Optics. 
I  1".    •  -plierical  .Vstrouomy. 

it      .'I       s  Inorganic  Uiiemistry. 
I  Gitv'orvS  O'-.'anic  Chemi.stry. 
!  Glimcirs  C;  lenlo-:. 

'^::i  -I  h's  A nf«l,>  rical  Geometry. 

■';■    -'lie's  Uoads  and  Hall  Koads. 


L  r.-rD  HIGHER  MATHEMATICS 
i»av\e.-;  EU'.m.-nts  of  SiirvevlnL'. 
l/,f.  ies'  Dlciionary  of  Matiiematics. 
'>,ivif>*  Analytical  Geometry. 
\>\\'.  s"  Calculus, 
i'aviis'  Descriptive  Geometry. 
i)avi<:"  Slia<les  and  Shadows. 

6.    HISTORY  An'D   infTHOLOGY. 

Willard's  School  HLst.  of  United  Star    .ID         .is  'dythology  (large.) 
Wlliard  s  Larger  Hist,  of  United  Stat.   .     D  •  '  •  it  s  .Mytn.ilo-.'v  for  Schools. 
Willard's  Universal  Hist. In  Perspective.  I  \  .     ■  "Jistory  of  Ancient  Hebrews. 
Gould's  Ali.son'8  Eirope.  I  ^  ^i  aidsLast  Leaves  of  Am   History. 

7.    ELOCUTION,  INTELLECTUAL  /^IILOSOPHY,  RHETORIC,  ET^ 

Northends  Little  S(»('aker.  ■  '^':    •     >  Intellectual  Philosophy. 

Northends  American  Sp.-aker.  !  •    •   d  .-.  i'.ames'  EJomeiits  of  Crilicism. 

Northend's  School  Dialogues. 
Zachos'  Nf  nerican  Speaker. 

Bovd's  Lo;.        >r  Schools. 


Mah 


for  Colleges. 


Pc^  .1  1 


Art  of  Phetwric. 
rd  H  Moral  Philosophy,  in  press. 
-  on  the  Mind, 


Honier  s  JUlad. 


